We carry out the proof via energy analysis of suitable Galerkin approximations in space. We refer to, for example, [1]}, [2]} for general details on the method and [3]} for its use in the analysis of nonlinear acoustic models. Compared to similar energy arguments in the literature, the challenge here is to arrive at a bound that does not degenerate as sound diffusivity approaches zero.
We note that \(p \in X^{\textup {W}}\)  implies
\(\begin{aligned}p \in C_w([0,T]; {H_\diamondsuit ^3(\Omega )}), \qquad p_t\in C_w([0,T]; {H_\diamondsuit ^2(\Omega )}),\end{aligned}\) 
and so the initial conditions are meaningful.
Semi-discretization in space
Take \(\lbrace w_i\rbrace _{i \in \mathbb {N}}\)  to be the complete set of (smooth) eigenfunctions of \(-\Delta \)  on \(H_0^1(\Omega )\) . Fix \(n \in \mathbb {N}\)  and denote \(V_n=\text{span}\lbrace w_1, \ldots , w_n\rbrace \) . We seek an approximate solution in the form of
\(\begin{aligned}p^n=& \, \displaystyle \sum _{i=1}^n \xi _i(t)w_i(x),\\\end{aligned}\) 
where the unknown coefficients are \(\xi _i:(0,T) \rightarrow \mathbb {R}\) , \(i \in [1,n]\) . We choose the approximate initial data
\(\begin{aligned}(p^n(0), p^n_t(0))=(p^n_0, p^n_1) \in V_n \times V_n\end{aligned}\) 
such that
\(\begin{aligned}& p^n_0 \rightarrow p_0 \quad \text{ in } {H_\diamondsuit ^3(\Omega )}, \\& p^n_1 \rightarrow p_1 \quad \text{ in } {H_\diamondsuit ^2(\Omega )},\end{aligned}\) 
which is possible since \(V_n\)  is dense in \({H_\diamondsuit ^3(\Omega )}\) . We then study the following semi-discrete approximation of our problem:
\( \begin{aligned}{\left\lbrace \begin{array}{ll}(p_{tt}^n - \tfrac{b}{\alpha } p̥_t^n - \tfrac{c^2}{\alpha }p̥^n - \tfrac{k}{\alpha } q_t p_t^n, \phi )_{L^2} = 0,\\[1mm]\text{for every $\phi \in V_n$ pointwise a.e. in $(0,T)$}, \\[1mm](p^n(0), p_t^n(0))=(p^n_0, p^n_1).\end{array}\right.}\end{aligned}\) 
To arrive at a matrix equation, we also introduce matrices \(M^n=[M_{ij}]\) , \(K^n=[K_{ij}]\)  and \(C^n=[C_{ij}]\)  as
\(\begin{aligned}& M^n_{ij}=( w_i, w_j)_{L^2}, \\& K^n_{ij}(t)=-(\tfrac{c^2}{\alpha (t)}\Delta w_i, w_j)_{L^2},\quad C^n_{ij}(t)=-(\tfrac{b}{\alpha (t)}\Delta w_i, w_j)_{L^2}-(\tfrac{k}{\alpha (t)}q_t(t) w_i, w_j)_{L^2},\\\end{aligned}\) 
for \(i,j \in [1,n]\) . By setting \(\xi ^n=[\xi _{1} \ldots \xi _{n}]^T\) , \(\xi _0^n=[\xi _{1, 0} \ldots \xi _{n, 0}]^T\) , and \(\xi _1^n=[\xi _{1, 1} \ldots \xi _{n,n}]^T\) , problem (REF ) can be further rewritten as
\( \begin{aligned}{\left\lbrace \begin{array}{ll}M^n \xi ^n_{tt}+C^n(t) \xi _t^n+K^n(t)\xi ^n=0, \\(\xi ^n(0), \xi ^n_t(0))=(\xi ^n_0, \xi ^n_1).\end{array}\right.}\end{aligned}\) 
After additionally rewriting (REF ) as a first-order system and owing to the fact that the matrix \(M^n\)  is positive definite, existence of an absolutely continuous solution \([\xi ^n, \xi ^n_t]^{T}\)  on \([0, T_n]\)  for some \(T_n \le T\)  follows by ODE existence theory; see, for example, [2]}. To see that \(\xi ^n \in H^2(0, T_n)\) , we can employ a standard bootstrap argument; cf. [3]}. We thus conclude that problem (REF ) has a solution \(p^n \in H^2(0,T_n; V_n) \subset C^1([0,T_n]; V_n)\) .

A priori estimates
Our next goal is to derive a uniform in \(n\)  estimate for the solution of (REF ). Keeping in mind that we wish to arrive at a higher-order bound, we test the semi-discrete problem with \(\phi =\Delta ^2 p^n_t \in V_n\)  to arrive at
\( \begin{aligned}(p_{tt}^n - \tfrac{b}{\alpha } \Delta p_t^n -\tfrac{c^2}{\alpha }\Delta p^n - \tfrac{k}{\alpha } q_t p_t^n, \Delta ^2 p^n_t)_{L^2} = 0,\end{aligned}\) 
a.e. in time. Whenever the temporal argument is skipped here and in the following, we suppose the stated (in)equality to hold for almost every time instance in \((0,T_n)\) . We have chosen our discrete space so that \(p_{tt}^n= \Delta p^n_t=0\)  on \(\partial \Omega \)  and, therefore, we have
\(\begin{aligned}-(\alpha ^{-1}p̥^n, 2 p_t^n)_{L^2} =&\, (p̥^n \nabla [\alpha ^{-1}] + \alpha ^{-1}\nabla p̥^n, \nabla )_{L^2}\\=&\,\begin{multlined}[t] (p̥^n \nabla [\alpha ^{-1}], \nabla )_{L^2}\\+\frac{1}{2}\frac{\textup {d}}{\textup {d}t}(\alpha ^{-1}\nabla p̥^n, \nabla p̥^n)_{L^2} -\frac{1}{2} ([\alpha ^{-1}]_t \nabla p̥^n, \nabla p̥^n)_{L^2}\end{multlined}\\=&\,\begin{multlined}[t] -(\nabla p̥^n\cdot \nabla [\alpha ^{-1}] + p̥^n \alpha ^{-1}], )_{L^2}\\+\frac{1}{2}\frac{\textup {d}}{\textup {d}t}(\alpha ^{-1}\nabla p̥^n, \nabla p̥^n)_{L^2}-\frac{1}{2} ([\alpha ^{-1}]_t \nabla p̥^n, \nabla p̥^n)_{L^2}.\end{multlined}\end{aligned}\) 
Furthermore, it holds
\(\begin{aligned}\alpha ^{-1}q_t\, p_t^n] &= 2\alpha ^{-1}\nabla q_t\cdot \nabla p_t^n+ \alpha ^{-1}q_t + \alpha ^{-1}p_t^nq̥_t\\&\qquad +\alpha ^{-1}] \, q_t p_t^n+ 2\nabla [\alpha ^{-1}] \cdot \nabla q_t\, p_t^n+ 2\nabla [\alpha ^{-1}] \cdot \nabla p_t^n\, q_t.\end{aligned}\) 
Thus, we arrive at the following energy identity:
\( \begin{aligned}&\frac{1}{2}\frac{\textup {d}}{\textup {d}t}\Vert \Vert _{L^2}^2 + b \Vert \sqrt{\alpha ^{-1}}\nabla \Vert _{L^2}^2 + c^2 \frac{1}{2}\frac{\textup {d}}{\textup {d}t}\Vert \sqrt{\alpha ^{-1}}\nabla p̥^n\Vert _{L^2}^2 \\&=k (2\alpha ^{-1}\nabla q_t\cdot \nabla p_t^n+ \alpha ^{-1}q_t + \alpha ^{-1}p_t^nq̥_t+ 2\nabla [\alpha ^{-1}] \cdot \nabla p_t^n\, q_t + 2\nabla [\alpha ^{-1}] \cdot \nabla q_t\, p_t^n\\&\qquad +\alpha ^{-1}] \, q_t p_t^n,  )_{L^2}\\&\quad - b (\nabla [\alpha ^{-1}], \nabla )_{L^2}+ c^2 (\nabla p̥^n\cdot \nabla [\alpha ^{-1}] + p̥^n \alpha ^{-1}], )_{L^2}\\&\quad + c^2\frac{1}{2} ([\alpha ^{-1}]_t \nabla p̥^n, \nabla p̥^n)_{L^2}.\end{aligned}\) 
We can rely on the following estimates to handle the \(\alpha \)  terms on the right:
\(\begin{aligned}\Vert \alpha ^{-1}\Vert _{L^\infty }\le &\, \tfrac{1}{\underline{\alpha }}, \\\Vert \nabla [\alpha ^{-1}]\Vert _{L^\infty }=&\,\Vert \alpha ^{-2}\nabla \alpha \Vert _{L^\infty }\le \tfrac{1}{\underline{\alpha }^2} |k| C_{H^2, L^\infty }\Vert \nabla q̥\Vert _{L^2}=:\tilde{C}_1\,|k|\Vert \nabla q̥\Vert _{L^2}\\\Vert \alpha ^{-1}]\Vert _{L^4}=&\,\Vert -2\alpha ^{-3}|\nabla \alpha |^2 + \alpha ^{-2}\Vert _{L^4},\\\le &\, \tfrac{2}{\underline{\alpha }^3} |k|^2 C_{H^2, L^8}\Vert \nabla q̥\Vert _{L^2}^2+ \tfrac{1}{\underline{\alpha }^2} |k| C_{H^1, L^4}\Vert \nabla q̥\Vert _{L^2}\\=&\,: |k|\Vert \nabla q̥\Vert _{L^2}(\tilde{C}_2+\tilde{C}_3\,|k|\Vert \nabla q̥\Vert _{L^2})\end{aligned}\) 
as well as
\(\begin{aligned}\Vert [\alpha ^{-1}]_t\Vert _{L^\infty }&=\Vert \alpha ^{-2}\alpha _t\Vert _{L^\infty }\le \tfrac{1}{\underline{\alpha }^2} |k| C_{H^2, L^\infty }\Vert q̥_t\Vert _{L^2} =:\tilde{C}_4 |k| \Vert q̥_t\Vert _{L^2}.\end{aligned}\) 
Combined with Young's inequality, this approach yields the energy estimate
\(\begin{aligned}&\frac{1}{2}\frac{\textup {d}}{\textup {d}t}\Vert \Vert _{L^2}^2 + b \frac{1}{\overline{\alpha }}\,\Vert \nabla \Vert _{L^2}^2 + c^2 \frac{1}{\overline{\alpha }}\,\frac{1}{2}\frac{\textup {d}}{\textup {d}t}\Vert \nabla \Vert _{L^2}^2 \\\le &\, \begin{multlined}[t]\left(2\tfrac{|k|}{\underline{\alpha }} ((C_{H^1, L^4})^2+C_{H^2, L^\infty }) \right. \\ \left.+ |k|\Vert \nabla q̥\Vert _{L^2} \left(4 \tilde{C}_1 C_{\textup {PF}}C_{H^2, L^\infty }+ \bigl (\tilde{C}_2+\tilde{C}_3\,|k|\Vert \nabla q̥\Vert _{L^2}\bigr )C_{H^2, L^8}\right) \right)\\\times \Vert q̥_t\Vert _{L^2}\, \Vert \Vert _{L^2}^2\\+ \frac{b\overline{\alpha }}{2} \tilde{C}_1^2\,|k|^2\Vert \nabla q̥\Vert _{L^2}^2 \Vert \Vert _{L^2}^2+ \frac{b}{2\overline{\alpha }}\Vert \nabla \Vert _{L^2}^2\\+ c^2|k|\Vert \nabla q̥\Vert _{L^2} \bigl (\tilde{C}_1+(\tilde{C}_2+\tilde{C}_3\,|k|\Vert \nabla q̥\Vert _{L^2}) C_{H^1, L^4}\bigr )\\\times \Vert \nabla \Vert _{L^2}\, \Vert \Vert _{L^2}\\+ c^2 \tilde{C}_4 |k| \Vert q̥_t\Vert _{L^2} \Vert \nabla \Vert _{L^2}^2. \end{multlined}\end{aligned}\) 
We next estimate \(\Vert \nabla p_{tt}^n\Vert _{L^2}\) . Since \(p^n\)  satisfies the semi-discrete equation, we have
\(\begin{aligned}\Vert \nabla p_{tt}^n\Vert _{L^2}^2=&\, (\nabla p_{tt}^n, \nabla p_{tt}^n)_{L^2}=-(p_{tt}^n, \Delta p_{tt}^n)_{L^2}\\=&\, -(\alpha ^{-1}(b \Delta p_t^n +c^2\Delta p^n + k q_t p_t^n), \Delta p_{tt}^n)_{L^2}.\end{aligned}\) 
Therefore,
\( \begin{aligned}\Vert \nabla p_{tt}^n\Vert _{L^2}^2=&\, \begin{multlined}[t](\alpha ^{-1}(b \nabla \Delta p_t^n +c^2 \nabla \Delta p^n + k \nabla [q_t p_t^n]), \nabla p_{tt}^n)_{L^2} \\[1mm]+(\nabla [\alpha ^{-1}](b \Delta p_t^n +c^2 \Delta p^n + k q_t p_t^n), \nabla p_{tt}^n)_{L^2}. \end{multlined}\end{aligned}\) 
The above identity together with the Cauchy–Schwarz inequality allow us to conclude that
\(\begin{aligned}&\Vert \nabla p_{tt}^n\Vert _{L^2}\\\le &\, \begin{multlined}[t]\underline{\alpha }^{-1} (b\Vert \nabla \Delta p_t^n\Vert _{L^2} +c^2 \Vert \nabla \Delta p^n\Vert _{L^2} +| k| \Vert \nabla q_t\Vert _{L^2}C_{H^2, L^\infty }\Vert p̥_t^n\Vert _{L^2}\\+ |k|C_{H^1, L^4}^2 \Vert \nabla q_t\Vert _{L^2} \Vert \Delta p_t^n\Vert _{L^2})\\+\underline{\alpha }^{-2}|k|C_{H^2, L^\infty }\Vert \nabla \Delta q\Vert _{L^2} (b \Vert \Delta p_t^n\Vert _{L^2} +c^2 \Vert \Delta p^n\Vert _{L^2}\\ + |k |\Vert q_t\Vert _{L^2}C_{H^2, L^\infty }\Vert p̥_t^n\Vert _{L^2}).\end{multlined}\end{aligned}\) 
Applying Gronwall's inequality to (REF ) and taking the supremum over \(t \in (0,T_n)\)  yield
\(\begin{aligned}& \begin{multlined}[t] \sup _{t \in (0,T_n)} \Vert (t)\Vert _{L^2}^2\,+\sup _{t \in (0,T_n)} \Vert \nabla (t)\Vert _{L^2}^2+b \int _0^{T_n} \Vert \nabla \Vert _{L^2}^2\, \textup {d} t\end{multlined}\\\le &\, \begin{multlined}[t] C(T)(\Vert \nabla p̥_0\Vert _{L^2}^2+\Vert p̥_1\Vert _{L^2}^2),\end{multlined}\end{aligned}\) 
where we have additionally bounded the approximate initial data by the exact data in the same norm. The constant above is given by
\(\begin{aligned}C(T)= C_1\exp (C_2\Vert k|(\Vert q\Vert _{X^W_\alpha }+(1+b^2)\Vert q\Vert _{X^W_\alpha }^2+\Vert q\Vert _{X^W_\alpha }^3) T),\end{aligned}\) 
where \(C_1\) , \(C_2>0\)  do not depend neither on \(b\)  nor on
\(T_n\) ,
\(n\) . Thanks to (REF ), we then also have a uniform bound on \(p_{tt}^n\) :
\(\begin{aligned}\int _0^{T_n} \Vert \nabla p_{tt}^n\Vert _{L^2}^2 \, \textup {d} t\lesssim &\, \begin{multlined}[t]b^2\Vert \nabla \Delta p_t^n\Vert _{L^2 (L^2)} + \Vert \nabla \Delta p^n\Vert _{L^2 (L^2)}^2\\+ (1+\Vert \nabla q_t\Vert _{L^\infty (L^2)}^2) \Vert \Delta p_t^n\Vert _{L^2 (L^2)}^2\\+\Vert \nabla \Delta q\Vert _{L^\infty (L^2)}^2 \left(b^2 \Vert \Delta p_t^n\Vert _{L^2 (L^2)}^2 + \Vert \Delta p^n\Vert _{L^2 (L^2)}^2 \right.\\ \left.+ \Vert q_t\Vert _{L^\infty (L^2)}^2 \Vert p̥_t^n\Vert _{L^2 (L^2)}^2 \right).\end{multlined}\end{aligned}\) 
If \(b=0\) , we even get a bound on \(\sup _{(0,T_n)} \Vert \nabla p_{tt}^n\Vert _{L^2}\)  here.
The uniform bounds with respect to \(T_n\)  allow us to extend the existence interval to \([0,T]\) .

Passing to the limit
Thanks to the uniform bounds on \(p^n\)  and the fact that our spatial and temporal domains are bounded, we can employ standard compactness arguments and conclude that there exist a subsequence, which we do not relabel, and a function \(p\)  such that
\( \begin{alignedat}{4}p_{tt}^n &\rightharpoonup p_{tt} &&\text{ weakly} &&\text{ in } &&L^2(0,T;H_0^1(\Omega )), \\p_t^n &\rightharpoonup p_t &&\text{ weakly-$\star $} &&\text{ in } &&L^\infty (0,T; H_0^1(\Omega )\cap H^2(\Omega )),\\p_t^n &\rightharpoonup p_t &&\text{ weakly} &&\text{ in } &&L^2(0,T; {H_\diamondsuit ^3(\Omega )})\ \mbox{ if }b>0,\\p^n &\rightharpoonup p &&\text{ weakly-$\star $} &&\text{ in } &&L^\infty (0,T; {H_\diamondsuit ^3(\Omega )}).\end{alignedat}\) 
We next wish to prove that this limit solves (REF ). We multiply the semi-discrete equation by \(\eta \in C_0^\infty (0,T)\)  and integrate over time to obtain
\( \begin{aligned}\int _{0}^T(p^n_{tt}, w_i)_{L^2}\, \eta (t)\, \textup {d}t=\, \int _{0}^T(c^2\Delta p^n+b\Delta p_t^n+kq_t p_t^n, \alpha ^{-1}w_i)_{L^2} \eta (t) \, \textup {d}t\end{aligned}\) 
for \(i \in [1, n]\) . Letting \(n \rightarrow \infty \)  in the above equation and employing (REF ) leads to
\(\begin{aligned}\int _{0}^T(p_{tt}, w_i)_{L^2}\, \eta (t)\, \textup {d}t=\, \int _{0}^T(c^2\Delta p+b\Delta p_t+k q_t p_t, \alpha ^{-1}w_i)_{L^2} \eta (t) \, \textup {d}t,\end{aligned}\) 
for all \(i \in \mathbb {N}\)  and \(\eta \in C^\infty _0(0,T)\) . Since \(\displaystyle \bigcup _{n \in \mathbb {N}}V_n\)  is dense in \(L^2(\Omega )\) , and by the Fundamental Lemma of Calculus of Variations, \(p\)  solves the original problem. For \(b \ne 0\) , due to the embedding
\(\begin{aligned}& p^n \in H^{1}(0,T; {H_\diamondsuit ^3(\Omega )}) \hookrightarrow C([0,T]; {H_\diamondsuit ^3(\Omega )}),\end{aligned}\) 
we know that \(p^n(0) \rightharpoonup p(0) \quad \text{in } {H_\diamondsuit ^3(\Omega )}\) . On the other hand, \(p^n(0) \rightarrow p_0\)  in \({H_\diamondsuit ^3(\Omega )}\) , and so \(p(0)=p_0\) . We also have
\(\begin{aligned}p^n_t \in &\, \begin{multlined}[t] L^\infty (0,T; {H_\diamondsuit ^2(\Omega )}) \cap H^1(0,T; H_0^1(\Omega )) \\ \hookrightarrow C_w([0,T]; {H_\diamondsuit ^2(\Omega )}) \cap C([0,T]; H_0^1(\Omega )),\end{multlined}\end{aligned}\) 
and, therefore, also \(p_t(0)=p_1\) .

Energy estimate for \(p\)
We can take the limit inferior as \(n \rightarrow \infty \)  of the discrete energy bound, and by virtue of the weak and the weak-\(\star \)  lower semi-continuity of norms obtain the final estimate. Uniqueness of a solution in \(X^{\textup {W}}\)  follows by testing the homogeneous problem (i.e., with \(p_0=p_1=0)\)  with \(p_t\)  and relying on the bound
\( \begin{aligned}&\frac{1}{2} \Vert \sqrt{\alpha (t)}p_t(t)\Vert ^2_{L^2}+c^2\Vert \nabla p(t)\Vert ^2_{L^2} +b \int _0^t\Vert \nabla p_t\Vert ^2\, \textup {d}x\textup {d}s\\=&\frac{1}{2} \int _0^t(\alpha _t p_t, p_t)_{L^2}\, \textup {d} s +k\int _0^t (q_t p_t, p_t)\, \textup {d} s \\\lesssim &\, (\Vert \alpha _t\Vert _{L^\infty (L^\infty )}+\Vert q_t\Vert _{L^\infty (L^\infty )})\Vert p_t\Vert ^2_{L^2(L^2)},\end{aligned}\) 
for \(t \in [0,T]\) , together with Gronwall's inequality.

Additional regularity
The already established existence of the solution \(p\in X^{\textup {W}}\) , via the PDE implies
\(p_{tt}=\alpha ^{-1}(b+c^2p̥+kq_tp_t)\in H^1(0,T;H^{-1}(\Omega ))\) 
and, therefore, \(p\in H^3(0,T;H^{-1}(\Omega ))\cap X^{\textup {W}}\) .
Since by interpolation (with interpolation parameter \(\theta =\frac{1-m}{2-m}\in [0,\frac{1}{2})\) , where \(m=\min \lbrace s,1\rbrace >0\)  and \(r=\frac{m}{2(2-m)}>0\) ) we have
\(\begin{aligned}H^3(0,T;H^s(\Omega ))\cap H^2(0,T;{H_\diamondsuit ^2(\Omega )})&\hookrightarrow H^{2+(1-\theta )}(0,T;H^{m+\theta (2-m)}(\Omega ))\\&=H^{5/2+r}(0,T;H^1(\Omega ))\hookrightarrow C^2([0,T];H^1(\Omega )),\end{aligned}\) 
the imposed initial conditions on \(p\) , \(p_t\) , and \(p_{tt}\)  make sense in the spaces \(H^3(\Omega )\) , \(H^3(\Omega )\) , and \(H^1(\Omega )\) , respectively.
We now derive a uniform bound on \(\Vert \nabla p_{tt}\Vert _{L^\infty (L^2)}\)  also in the case \(b>0\)  for sufficiently regular initial data. Note that we have already established such a bound when \(b=0\) , right after estimate (REF ).
To justify the derivation of this bound, we will employ a Galerkin discretization of the time-differentiated version of (REF ):
\(\begin{aligned}{\left\lbrace \begin{array}{ll}(\alpha \tilde{p}^n_{ttt} +\alpha _t \tilde{p}^n_{tt}- b {p}^n_{tt} - c^2{p}^n_t - k q_t \tilde{p}^n_{tt} - {k q_{tt}\tilde{p}^n_t}, \phi )_{L^2} = 0,\\[1mm]\text{for every $\phi \in V_n$ pointwise a.e. in $(0,T)$}, \\[1mm](\tilde{p}^n(0), \tilde{p}^n_t(0), \tilde{p}^n_{tt}(0))=(\tilde{p}^n_0, \tilde{p}^n_1, \tilde{p}^n_2),\end{array}\right.}\end{aligned}\) 
where \(\tilde{p}^n_0\)  , \(\tilde{p}^n_1\) , \(\tilde{p}^n_2\)  are the Galerkin projections of \(p_0\) , \(p_1\) , and \(\alpha (0)^{-1} (b p̥_1 +c^2p̥_0 +k q_t(0)p_1)\)  (for the latter see the compatibility condition (REF )). To relate the solution of the time-differentiated equation to the original one, the compatibility condition (REF ) is needed.
Testing
(REF )
with \(\tilde{p}^n_{ttt}-\Delta \tilde{p}^n_{tt}\)  results in
\(\begin{aligned}(\alpha \tilde{p}^n_{ttt}+\alpha _t \tilde{p}^n_{tt} - b \Delta \tilde{p}^n_{tt} -c^2\Delta \tilde{p}^n_t - k q_t \tilde{p}^n_{tt}-k q_{tt}\tilde{p}^n_t, \tilde{p}^n_{ttt}-\Delta \tilde{p}^n_{tt})_{L^2} = 0.\end{aligned}\) 
We can then rely on the identity
\(\begin{aligned}(\alpha \tilde{p}^n_{ttt}, \tilde{p}^n_{ttt}-\Delta \tilde{p}^n_{tt})_{L^2}=&\,(\alpha \tilde{p}^n_{ttt}, \tilde{p}^n_{ttt})_{L^2}+(\nabla [\alpha \tilde{p}^n_{ttt}], \nabla \tilde{p}^n_{tt})_{L^2} \\=&\,(\alpha \tilde{p}^n_{ttt}, \tilde{p}^n_{ttt})_{L^2}+ (\alpha \nabla \tilde{p}^n_{ttt}+ \tilde{p}^n_{ttt}\nabla \alpha , \nabla \tilde{p}^n_{tt})_{L^2}\\=&\,\begin{multlined}[t](\alpha \tilde{p}^n_{ttt}, \tilde{p}^n_{ttt})_{L^2}+\frac{1}{2} \frac{\textup {d}}{\textup {d}t}(\alpha \nabla \tilde{p}^n_{tt}, \nabla \tilde{p}^n_{tt})\\ -\frac{1}{2} (\alpha _t \nabla \tilde{p}^n_{tt},\nabla \tilde{p}^n_{tt})+( \tilde{p}^n_{ttt} \nabla \alpha ,\nabla \tilde{p}^n_{tt}),\end{multlined}\end{aligned}\) 
and, similarly, the identity
\(\begin{aligned}(\alpha _t \tilde{p}^n_{tt}, \tilde{p}^n_{ttt}-\Delta \tilde{p}^n_{tt})_{L^2}=&\,(\alpha _t \tilde{p}^n_{tt}, \tilde{p}^n_{ttt})_{L^2}+(\nabla [\alpha _t \tilde{p}^n_{tt}], \nabla \tilde{p}^n_{tt})_{L^2} \\=&\,\begin{multlined}[t](\alpha _t \tilde{p}^n_{tt}, \tilde{p}^n_{ttt})_{L^2}+(\alpha _t \nabla \tilde{p}^n_{tt}, \nabla \tilde{p}^n_{tt})_{L^2}+(\tilde{p}^n_{tt} \nabla \alpha _t, \nabla \tilde{p}^n_{tt})_{L^2}.\end{multlined}\end{aligned}\) 
In this manner, we obtain
\( \begin{aligned}&\begin{multlined}[t]\Vert \sqrt{\alpha } \tilde{p}^n_{ttt}\Vert _{L^2}^2+\frac{1}{2} \frac{\textup {d}}{\textup {d}t}\Vert \sqrt{\alpha } \nabla \tilde{p}^n_{tt}\Vert _{L^2}^2\\+ \frac{1}{2} c^2 \frac{\textup {d}}{\textup {d}t}\Vert {p}^n_t\Vert _{L^2}^2+b \Vert {p}^n_{tt}\Vert _{L^2}^2+ \frac{1}{2} b \frac{\textup {d}}{\textup {d}t}\Vert \nabla \tilde{p}^n_{tt}\Vert _{L^2}^2\end{multlined}\\=&\, \begin{multlined}[t]\frac{1}{2} ( \alpha _t \nabla \tilde{p}^n_{tt}, \nabla \tilde{p}^n_{tt})_{L^2}-(\tilde{p}^n_{ttt} \nabla \alpha , \nabla \tilde{p}^n_{tt})_{L^2}-(\alpha _t \tilde{p}^n_{tt}, \tilde{p}^n_{ttt})_{L^2}\\-(\tilde{p}^n_{tt} \nabla \alpha _t, \nabla \tilde{p}^n_{tt})_{L^2}+ k(q_t \tilde{p}^n_{tt}+ q_{tt}\tilde{p}^n_t, \tilde{p}^n_{ttt}-\Delta \tilde{p}^n_{tt})_{L^2}+c^2({p}^n_t, \tilde{p}^n_{ttt})_{L^2} =: \textup {rhs}. \end{multlined}\end{aligned}\) 
Furthermore, we have
\(\begin{aligned}(q_t \tilde{p}^n_{tt}+ q_{tt}\tilde{p}^n_t, \tilde{p}^n_{ttt}-\Delta \tilde{p}^n_{tt})_{L^2}=&\, (q_t \tilde{p}^n_{tt}+ q_{tt}\tilde{p}^n_t, \tilde{p}^n_{ttt})_{L^2}+(\nabla (q_t \tilde{p}^n_{tt}+ q_{tt}\tilde{p}^n_t), \nabla \tilde{p}^n_{tt})_{L^2}. \\\end{aligned}\) 
Thus, we can estimate the right-hand side of (REF ) as follows:
\(\begin{aligned}|\textup {rhs}| \lesssim &\, \begin{multlined}[t] (1+\Vert \alpha _t\Vert _{L^\infty })\Vert \nabla \tilde{p}^n_{tt}\Vert _{L^2}^2+\varepsilon \Vert \tilde{p}^n_{ttt}\Vert _{L^2}^2+(1+\Vert \nabla \alpha \Vert _{L^\infty }^2+\Vert q_t\Vert _{L^\infty })\Vert \nabla \tilde{p}^n_{tt}\Vert _{L^2}^2\\+ (\Vert \alpha _t\Vert _{L^3}^2+\Vert \nabla \alpha _t\Vert _{L^3}^2+\Vert q_t\Vert _{L^3}^2+\Vert \nabla q_t\Vert _{L^3}^2)\Vert \tilde{p}^n_{tt}\Vert _{L^6}^2 \\+\Vert q_{tt}\Vert _{L^3}^2\Vert \nabla \tilde{p}^n_t\Vert _{L^6}^2+\Vert q_{tt}\Vert _{L^3}^2\Vert \tilde{p}^n_t\Vert _{L^6}^2+\Vert \nabla q_{tt}\Vert _{L^2}^2\Vert \tilde{p}^n_t\Vert _{L^\infty }^2+\Vert {p}^n_t\Vert _{L^2}^2 \end{multlined}\end{aligned}\) 
for all \(t \in [0,T]\) , with \(\varepsilon \)  small enough, but independent of \(b\) . From the compatibility condition (REF ) and boundedness of the projection operator we can obtain a bound on \(\nabla \tilde{p}^n_{tt}(0)\)  as follows:
\(\begin{aligned}\Vert \nabla \tilde{p}^n_{tt}(0)\Vert _{L^2} \lesssim &\, \begin{multlined}[t] b\Vert \nabla \Delta p_1\Vert _{L^2} + \Vert \nabla \Delta p_0\Vert _{L^2} + \Vert \nabla q_t(0)\Vert _{L^2} \Vert p̥_1\Vert _{L^2}+ \Vert \nabla q_t(0)\Vert _{L^2} \Vert \Delta p_1\Vert _{L^2}\\+\Vert \nabla \Delta q(0)\Vert _{L^2} (b \Vert \Delta p_1\Vert _{L^2} +\Vert \Delta p_0\Vert _{L^2}+ \Vert q_t(0)\Vert _{L^2} \Vert p̥_1\Vert _{L^2}).\end{multlined}\end{aligned}\) 
Applying Gronwall's inequality thus yields the energy estimate
\( \begin{aligned}&\begin{multlined}[t]\Vert \sqrt{\alpha } \tilde{p}^n_{ttt}\Vert _{L^2}^2+\frac{1}{2} \frac{\textup {d}}{\textup {d}t}\Vert \sqrt{\alpha } \nabla \tilde{p}^n_{tt}\Vert _{L^2}^2\\+ \frac{1}{2} c^2 \frac{\textup {d}}{\textup {d}t}\Vert {p}^n_t\Vert _{L^2}^2+b \Vert {p}^n_{tt}\Vert _{L^2}^2+ \frac{1}{2} b \frac{\textup {d}}{\textup {d}t}\Vert \nabla \tilde{p}^n_{tt}\Vert _{L^2}^2\end{multlined}\\\lesssim & \begin{multlined}[t] \Vert \nabla p̥_0\Vert _{L^2}^2+\Vert p̥_1\Vert _{L^2}^2+b \Vert \nabla p̥_1\Vert _{L^2}^2,\end{multlined}\end{aligned}\) 
where we have used that also (REF ) holds.
Due to the uniform bound (REF ), \(\tilde{p}^n\)  has a weakly convergent subsequence whose limit \(\tilde{p}\)  inherits this bound. Moreover, \(\tilde{p}\)  solves the time-differentiated PDE and satisfies the compatibility condition (REF ). By integration with respect to time, it therefore satisfies the original initial-value problem (REF ) and thus by uniqueness has to coincide with \(p\) .
Combined with the previous uniform estimate for smooth approximations, this allows us to obtain the higher-order in time bound
(REF ).

Uniform bounds for the Westervelt equation
We next analyze the Westervelt equation by introducing the fixed-point mapping
\(\mathcal {T}:q\mapsto p,\) 
where \(q\)  will belong to a suitably chosen ball in the space
\( X^{\textup {W}}\cap L^\infty (0,T; L^\infty (\Omega )), \) 
with \(X^{\textup {W}}\)  defined in (REF ). The function \(p\)  will solve the linearized Westervelt equation:
\( \alpha p_{tt}- b p̥_t - c^2p̥ - k q_t p_t =0,\) 
with \(\alpha =1-kq\)  and initial conditions \(p(0)=q(0)=p_0\) , \(p_t(0)=q_t(0)=p_1\) .
We first determine the conditions under which \(\mathcal {T}\)  is a self-mapping on an appropriately chosen set \(M\) . More precisely, we define \(M\)  as a subset of the solution space \(X^{\textup {W}}\) ,
\(\begin{aligned}M = \left\lbrace \vphantom{\frac{1}{2|k|}} q\in X^{\textup {W}}:\right.&\, q(0)=p_0, \ q_t(0)=p_1, \ \Vert q\Vert _{L^\infty (L^\infty )}\le \frac{1}{2|k|} \\& \hspace*{-36.98866pt}\left. \Vert \nabla q_{tt}\Vert _{L^2 (L^2)}^2+\Vert q̥_t\Vert _{L^\infty (L^2)}^2 + \Vert \nabla q̥\Vert _{L^\infty (L^2)}^2+b\Vert \nabla q̥_t\Vert _{L^2 (L^2)}^2\le R^2 \vphantom{\frac{1}{2|k|}} \right\rbrace .\end{aligned}\) 
The imposed conditions allow us to uniformly bound the hidden constant in estimate (REF ) by
\(C_{\textup {lin}}(T,R)= C_1 \exp {\left(C_2|k|(R+R^2+R^3)T\right)},\) 
where the positive constants \(C_1\)  and \(C_2\)  do not depend on \(b \in [0, \bar{b})\)  for fixed \(\bar{b}\)  nor on \(R\) , \(T\) ; cf. (REF ).
Proposition 4.1 
Let \(\Omega \subset \mathbb {R}^n\) , where \(n \in \lbrace 1, 2, 3\rbrace \) , be bounded and \(C^3\)  regular. Furthermore, let \(b \in [0, \bar{b})\)  and let \(T>0\)  be a given final time. For initial data satisfying the regularity condition (REF ), the mapping \(\mathcal {T}:q\mapsto p\)  is a self-mapping on the set \(M\)  defined in (REF ), provided the \(H_0^1(\Omega ) \times L^2(\Omega )\)  norm of the initial data is small enough so that
\({C_{\textup {A}}\tilde{C}_1 \exp (\tilde{C}_2RT)(\Vert p_1\Vert ^2_{L^2}+\Vert \nabla p_0\Vert ^2_{L^2})^{1/4}R^{1/2} \le \frac{1}{2|k|}}\) 
and the radius \(R\)  and final time \(T\)  are chosen such that
\(C_1 \exp {\left(C_2|k|(R+R^2+R^3)T\right)}(\Vert \nabla p̥_0\Vert _{L^2}^2+\Vert p̥_1\Vert _{L^2}^2)\le R^2.\) 
Furthermore, \(p=\mathcal {T}(q)\)  satisfies the energy estimate
\(\begin{aligned}& \Vert \nabla p_{tt}\Vert _{L^2 (L^2)}^2+\operatornamewithlimits{ess\,sup}_{t \in (0,T)} \Vert (t)\Vert _{L^2}^2\,+\operatornamewithlimits{ess\,sup}_{t \in (0,T)} \Vert \nabla p̥(t)\Vert _{L^2}^2+b\Vert \nabla \Vert _{L^2 (L^2)}^2\\\le &\, C_{\textup {lin}}(T,R) (\Vert \nabla p̥_0\Vert _{L^2}^2+\Vert p̥_1\Vert _{L^2}^2).\end{aligned}\) 

Here the constants \(C_{\textup {A}}\) , \(\tilde{C}_1\) , \(\tilde{C}_2\) , \(C_1\) , \(C_2\) , and \(C_{\textup {lin}}(T,R)\)  are as in (REF ) and (REF ) below.

Remark 1 To satisfy condition (REF ), only smallness of the initial data in a weaker norm is needed; namely \(H_0^1(\Omega ) \times L^2(\Omega )\)  rather than the \({H_\diamondsuit ^3(\Omega )}\times {H_\diamondsuit ^2(\Omega )}\)  norm required for the higher-order energy estimates.
Condition (REF ) can be satisfied even for initial data with large higher-order norm, by, for example, choosing
\(R^2= 2 C_1 (\Vert \nabla p̥_0\Vert _{L^2}^2+\Vert p̥_1\Vert _{L^2}^2)\) 
and assuming the final time to be short enough, so that
\(T\le \frac{\ln (2)}{C_2|k|(R+R^2+R^3)}.\) 
Note that both smallness conditions (REF ) and (REF ) are additionally mitigated by the fact that the nonlinearity parameter \(k\)  is typically small in magnitude in ultrasonic applications; cf. [6]}.

We rely in the proof on our previous linear analysis, which, in particular, implies \(M\ne \emptyset \)  and that \(\mathcal {T}\)  is well-defined on \(M\) . Let \(q \in M\) . The bound
\(\Vert q\Vert _{L^\infty (L^\infty )}\le \frac{1}{2|k|}\) 
immediately implies that
the non-degeneracy assumption (REF ) holds with \(\underline{\alpha }=\frac{1}{2}\)  and \(\overline{\alpha }=\frac{3}{2}\) .
Moreover, the regularity assumptions of Proposition REF  are satisfied with the uniform bounds
\(\begin{aligned}&\Vert \alpha \Vert _{L^\infty (H^3)}\le |\Omega |^{1/2}+|k| C_{(-\Delta )^{-1}} R, \qquad \Vert \alpha _t\Vert _{L^\infty (H^2)}\le |k| C_{(-\Delta )^{-1}} R, \\&\Vert q_t\Vert _{L^\infty (H^2)}\le C_{(-\Delta )^{-1}} R\,.\end{aligned}\) 
Hence Proposition REF  applied to \(p=\mathcal {T}(q)\) , together with the imposed smallness condition
(REF ) imply that \(\mathcal {T}\)  is well-defined and yield the \(R^2\)  bound in (REF ) on \(\mathcal {T}q=p\) .
To also achieve the \(\frac{1}{2|k|}\)  bound on the \(L^\infty (0,T; L^\infty (\Omega ))\)  norm of \(p\) , we derive a simple energy estimate for the linearization.
Similarly to (REF ), by multiplying equation (REF ) with \(p_t\)  and integrating over space and time, we arrive at
\(\begin{aligned}&\frac{1}{2} \left\lbrace \Vert \sqrt{\alpha (t)}p_t(t)\Vert ^2_{L^2}+c^2\Vert \nabla p(t)\Vert ^2_{L^2} \right\rbrace \Big \vert _0^t+b \int _0^t\Vert \nabla p_t\Vert ^2\, \textup {d}x\textup {d}s\\\lesssim &\, (\Vert \alpha _t\Vert _{L^\infty (L^\infty )}+\Vert q_t\Vert _{L^\infty (L^\infty )})\Vert p_t\Vert ^2_{L^2(L^2)}.\end{aligned}\) 
From here by Gronwall's inequality, we have
\(\begin{aligned}&\Vert p_t(t)\Vert ^2_{L^2}+\Vert \nabla p(t)\Vert ^2_{L^2}\le \, C_1 \exp (C_2RT)(\Vert p_1\Vert ^2_{L^2}+\Vert \nabla p_0\Vert ^2_{L^2})\end{aligned}\) 
for all \(t \in [0,T]\) . Following, e.g., [7]}, the desired \(L^\infty \)  bound on \(p\)  can be obtained by employing the above estimate and Agmon's interpolation inequality [8]}:
\( \Vert p(t)\Vert _{L^\infty } \le C_{\textup {A}} \Vert p(t)\Vert _{H^1}^{1/2}\Vert p(t)\Vert _{H^2}^{1/2}.\) 
Indeed, by also using that \(\Vert p\Vert _{L^\infty (H^2)} \lesssim R\) , we have
\(\begin{aligned}\Vert p\Vert _{L^\infty (L^\infty )}\le C_{\textup {A}}\tilde{C}_1 \exp (\tilde{C}_2RT)(\Vert p_1\Vert ^2_{L^2}+\Vert \nabla p_0\Vert ^2_{L^2})^{1/4}R^{1/2}.\end{aligned}\) 
for some positive constants \(\tilde{C}_1\)  and \(\tilde{C}_2\) , independent of \(b\) . Thus for \(\mathcal {T}(M) \subset M\)  to hold, we need to impose the following condition:
\(C_{\textup {A}}\tilde{C}_1 \exp (\tilde{C}_2RT)(\Vert p_1\Vert ^2_{L^2}+\Vert \nabla p_0\Vert ^2_{L^2})^{1/4}R^{1/2} \le \frac{1}{2|k|},\) 
as claimed.
We next prove that \(\mathcal {T}\)  is a contraction in a suitably chosen topology.
Proposition 4.2 
Let the assumptions on \(\Omega \)  from Proposition REF  be satisfied.
There exist constants \(C_5\) , \(C_6>0\)  depending only on the domain \(\Omega \)  and the constant \(c^2\) , but neither on \(T\)  nor on \(b\) , such that if either
\(C_5 \exp {(C_6 T)} k^2 \, (T+1)\, (\Vert p_0\Vert ^2_{H^3}+\Vert p_1\Vert ^2_{H^2}) < 1\) 
or \(p_1 \in {H_\diamondsuit ^3(\Omega )}\) , (REF ), \(q\) , \(\alpha \in X^{\textup {W}}\)  with (REF ), and
\(C_5 \exp {(C_6 T)} k^2 \, T\, (\Vert p_0\Vert ^2_{H^3}+\Vert p_1\Vert ^2_{H^3}) < 1,\) 
then the mapping \(\mathcal {T}\)  is a contraction on \(M\)  in the topology induced by
\( \Vert p\Vert ^2_{X}= \Vert p_{tt}\Vert _{L^2 (L^2)}^2+\Vert \nabla p_t\Vert _{L^\infty (L^2)}^2 + \Vert \Delta p\Vert _{L^\infty (L^2)}^2.\) 

Fulfillment of condition “(REF ) or (REF )" can be achieved by either imposing smallness on the initial data in \({H_\diamondsuit ^3(\Omega )}\times {H_\diamondsuit ^2(\Omega )}\)  for a fixed given final time \(T\) , or by imposing short enough final time \(T\)  for fixed given initial data in \({H_\diamondsuit ^3(\Omega )}\times {H_\diamondsuit ^3(\Omega )}\) . For the latter, in case of \(b=0\) , it suffices to have \({H_\diamondsuit ^3(\Omega )}\times {H_\diamondsuit ^2(\Omega )}\)  regular data.
In order to prove strict contractivity, we take \(q^{(1)}\)  and \(q^{(2)}\)  in \(M\) . Let then \(p^{(1)}=\mathcal {T} q^{(1)}\)  and \(p^{(2)}=\mathcal {T} q^{(2)} \) . Furthermore, we introduce the differences \(\overline{p}=p^{(1)} -p^{(2)} \)  and \(\overline{q}= q^{(1)} -q^{(2)} \) . Then we know that \(\overline{p}\)  solves the equation
\( (1-k q^{(1)} )\overline{p}_{tt}-c^2 \Delta \overline{p}-b \Delta \overline{p}_t=k q_t^{(1)}\overline{p}_t+\underbrace{k\overline{q}_tp^{(2)}_t+k \overline{q}p^{(2)} _{tt}}_{:=f}\) 
and has zero initial conditions.
Testing equation (REF ) with \(-\Delta \overline{p}_t\) , integrating over space and \((0,t)\) , and noting that \(\overline{p}(0)=\overline{p}_t(0)=0\)  yields
\(\begin{aligned}&\frac{1}{2}\Vert \nabla \overline{p}_t(t)\Vert _{L^2}^2+ \frac{c^2}{2} \Vert \Delta \overline{p}(t)\Vert _{L^2}^2+b \int _0^t \Vert _t\Vert _{L^2}^2 \, \textup {d} s \\=&\,\begin{multlined}[t] k \int _0^t \int _{\Omega }\nabla (q^{(1)}\overline{p}_{tt}) \cdot \nabla \overline{p}_t \, \textup {d} x\textup {d}s+ k \int _0^t \int _{\Omega }\nabla (q^{(1)}_t\overline{p}_{t}) \cdot \nabla \overline{p}_t \, \textup {d} x\textup {d}s\\+ \int _0^t \int _{\Omega }\nabla f \cdot \nabla \overline{p}_t \, \textup {d} x\textup {d}s \end{multlined}\\=&\, \,\begin{multlined}[t] \frac{1}{2} k \int _{\Omega }q^{(1)}(t) |\nabla \overline{p}_{t}(t)|^2\, \textup {d} x- \frac{1}{2} k \int _0^t \int _{\Omega }q^{(1)}_t |\nabla \overline{p}_{t}|^2 \, \textup {d} x\textup {d}s\\ +k \int _0^t \int _{\Omega }\overline{p}_{tt} \nabla q^{(1)} \cdot \nabla \overline{p}_t \, \textup {d} x\textup {d}s+ k \int _0^t \int _{\Omega }\nabla (q^{(1)}_t\overline{p}_{t}) \cdot \nabla \overline{p}_t \, \textup {d} x\textup {d}s\\+ \int _0^t \int _{\Omega }\nabla f \cdot \nabla \overline{p}_t \, \textup {d} x\textup {d}s. \end{multlined}\end{aligned}\) 
From here, we further have by Young's inequality
\( \begin{aligned}&\frac{1}{2}\underline{\alpha }\Vert \nabla \overline{p}_t(t)\Vert _{L^2}^2+ \frac{c^2}{2} \Vert \Delta \overline{p}(t)\Vert _{L^2}^2 +b \Vert \Delta \overline{p}_t\Vert _{L^2 (L^2)}^2 \\\le &\, \,\begin{multlined}[t] \frac{1}{2}|k| \Vert q^{(1)}_t\Vert _{L^\infty (L^\infty )} \Vert \nabla \overline{p}_{t}\Vert _{L^2 (L^2)}^2 + |k| \Vert \overline{p}_{tt}\Vert _{L^2 (L^2)} \Vert \nabla q^{(1)}\Vert _{L^\infty (L^\infty )} \Vert \nabla \overline{p}_t\Vert _{L^2 (L^2)}\\+ |k| \Vert \nabla q^{(1)}_t\Vert _{L^\infty (L^3)} \Vert \overline{p}_{t}\Vert _{L^2 (L^6)} \Vert \nabla \overline{p}_t \Vert _{L^2(L^2)}+ \Vert \nabla f\Vert _{L^2 (L^2)} \Vert \nabla \overline{p}_t\Vert _{L^2 (L^2)}. \end{multlined}\end{aligned}\) 
Note that we can derive a bound on the \(\nabla f\)  term as follows:
\(\begin{aligned}\Vert \nabla f\Vert _{L^2 (L^2)}^2=&\, k^2 \int _0^t \int _{\Omega }|\nabla (\overline{q}_t p^{(2)}_t+ \overline{q}p^{(2)} _{tt}) |^2 \, \textup {d} x\textup {d}s \\\le &\,\begin{multlined}[t] 2 k^2(\Vert \nabla \overline{q}_t\Vert _{L^\infty (L^2)}^2 \Vert p^{(2)}_t\Vert _{L^2 (L^\infty )}^2+\Vert \overline{q}_t\Vert _{L^\infty (L^3)}^2\Vert \nabla p^{(2)}_t\Vert _{L^2 (L^6)}^2\\+ \Vert \nabla \overline{q}\Vert _{L^\infty (L^3)}^2 \Vert p^{(2)}_{tt}\Vert _{L^2 (L^6)}^2+\Vert \overline{q}\Vert _{L^\infty (L^\infty )}^2\Vert \nabla p^{(2)}_{tt}\Vert _{L^2 (L^2)}^2 ).\end{multlined}\end{aligned}\) 
Testing (REF ) additionally with \({\overline{p}_{tt}}\)  produces
\(\begin{aligned}&\underline{\alpha }\Vert \overline{p}_{tt}\Vert ^2_{L^2 L^2} +\frac{b}{2}\Vert \nabla \overline{p}_t(t)\Vert _{L^2}^2\\\le &\, \left(c^2\Vert \Delta \overline{p}\Vert _{L^2 (L^2)}+|k| \Vert q^{(1)}_t\Vert _{L^\infty (L^3)}\Vert \overline{p}_{t}\Vert _{L^2(L^6)}+\Vert f\Vert _{L^2 (L^2)} \right)\Vert \overline{p}_{tt}\Vert _{L^2 (L^2)}.\end{aligned}\) 
Combining the estimates derived above and then employing Gronwall's inequality yields
\(\begin{aligned}& \Vert \overline{p}_{tt}\Vert _{L^2 (L^2)}^2 + \Vert \nabla \overline{p}_t(t)\Vert _{L^2}^2+ \Vert \Delta \overline{p}(t)\Vert _{L^2}^2+b \Vert _t\Vert _{L^2 (L^2)}^2+b\Vert \nabla \overline{p}_t(t)\Vert _{L^2}^2 \\\le &\, C_1 \exp {(C_2 T)} \Vert f\Vert ^2_{L^2 (H^1)},\end{aligned}\) 
where the positive constants \(C_1\)  and \(C_2\)  do not depend on \(b\) . From here we have
\( \begin{aligned}\Vert \overline{p}\Vert ^2_{X}\le &\, \begin{multlined}[t] C_3 \exp {(C_2 T)} k^2 (T \Vert  p^{(2)}_t\Vert ^2_{L^\infty (L^\infty )}+ T\Vert \nabla p^{(2)}_t\Vert ^2_{L^\infty (L^6)}\\+ \Vert  p^{(2)}_{tt}\Vert ^2_{L^2(L^6)}+\Vert \nabla p^{(2)}_{tt}\Vert ^2_{L^2(L^2)} )\Vert \overline{q}\Vert ^2_{X} \end{multlined}\\\le &\, C_4 \exp {(C_2 T)} k^2 (T+1) C_{\textup {lin}}(T)(\Vert p_0\Vert ^2_{H^3}+\Vert p_1\Vert ^2_{H^2}) \Vert \overline{q}\Vert ^2_{X},\\\le &\, C_5 \exp {(C_6 T)} k^2 (T+1) (\Vert p_0\Vert ^2_{H^3}+\Vert p_1\Vert ^2_{H^2}) \Vert \overline{q}\Vert ^2_{X}.\end{aligned}\) 
Alternatively, assuming the initial data to be smoother so that the estimate (REF ) applies, we can estimate \(\Vert  p^{(2)}_{tt}\Vert ^2_{L^2(L^6)}\)  by \(T\Vert  p^{(2)}_{tt}\Vert ^2_{L^\infty (L^6)}\)  and thus get
\(\Vert \overline{p}\Vert ^2_{X}\le \, C_5 \exp {(C_6 T)} k^2 \, T\, (\Vert p_0\Vert ^2_{H^3}+\Vert p_1\Vert ^2_{H^3}) \Vert \overline{q}\Vert ^2_{X},\) 
in place of (REF ).
Here \(C_3, \ldots , C_6>0\)  do not depend on \(b\) . We, therefore, conclude that \(\mathcal {T}\)  is strictly contractive provided the data is small enough so that
(REF ) or (REF ) holds.
The two previous results allow us to now employ a contraction principle and obtain a local well-posedness result for the Westervelt equation with a uniform bound in \(b\)  for small and smooth data.
Theorem 4.1 
Let \(\Omega \subset \mathbb {R}^n\) , where \(n \in \lbrace 1, 2, 3\rbrace \) , be bounded and \(C^3\)  regular. Let \(k \in \mathbb {R}\) . Furthermore, let \(b \in [0, \bar{b})\) . There exist constants \(C_1\) , \(C_2\) , \(C_5\) , \(C_6>0\)  that depend only on the domain \(\Omega \)  and the constant \(c^2\) , but neither on final time \(T\)  nor on \(b\) , such that if, for some \(\delta \) , \(R>0\) ,
\({C_{\textup {A}}\tilde{C}_1 \exp (\tilde{C}_2RT)\sqrt{\delta } R^{1/2} \le \frac{1}{2|k|},}\) 
as well as (REF ) or (REF ) hold,
then for any initial data satisfying (REF ), (REF ), and
\({\Vert  p_1\Vert _{L^2}^2+\Vert \nabla p_0\Vert _{L^2}^2 \le \delta ^2,}\) 
there exists a unique solution \(p\in X^{\textup {W}}\)  of problem
\( \left\lbrace \begin{aligned}p_{tt}- b p̥_t - c^2p̥ =&\, \frac{k}{2} (p^2)_{tt} &&\quad \text{ in }\Omega \times (0,T), \\[1mm]p=&\,0 &&\quad \text{ on } \partial \Omega \times (0,T),\\[1mm](p, p_t)=&\,(p_0, p_1) &&\quad \mbox{ in }\Omega \times \lbrace 0\rbrace ,\end{aligned} \right.\) 
where \(X^{\textup {W}}\)  is defined in (REF ). Furthermore, solution \(p\)  satisfies the estimate (REF ).

The proof follows by employing the Banach fixed-point theorem to the mapping
\(\mathcal {T}: M \ni q \mapsto p,\) 
with the ball \(M\)  defined in (REF ) and \(p\)  being the solution of
\((1-kq)p_{tt}- b p̥_t - c^2p̥ - k q_t p_t =0,\) 
with initial conditions \(p(0)=q(0)=p_0\) , \(p_t(0)=q_t(0)=p_1\) .
We first note that the space \((M, d)\)  with the metric \(d(x,y)=\Vert x-y\Vert _{X}\)  is a closed subset of a complete normed space, where we recall that
\(\Vert p\Vert ^2_{X}= \Vert p_{tt}\Vert _{L^2 (L^2)}^2+\Vert \nabla p_t\Vert _{L^\infty (L^2)}^2 + \Vert \Delta p\Vert _{L^\infty (L^2)}^2.\) 
 Indeed, \(M\)  is a ball in \(X^{\textup {W}}\cap L^\infty (0,T; L^\infty (\Omega ))\) , thus it is weakly-\(\star \)  closed in \(X^{\textup {W}}\)  by virtue of the Banach–Alaoglu theorem. Therefore, any Cauchy sequence with respect to the \(X\)  norm in \(M\)  converges to some
\(y\in L^\infty (0,T; {H_\diamondsuit ^2(\Omega )}) \cap W^{1, \infty }(0,T; H_0^1(\Omega )) \cap H^2(0,T; L^2(\Omega ))\) 
and has a weakly-\(\star \)  in \(X^{\textup {W}}\)  convergent subsequence with some limit \(x\in M\) . Due to uniqueness of limits we have \(y=x\in M\) .
Proposition REF  guarantees the \(\mathcal {T}\)  is well-defined self-mapping, whereas Proposition REF  yields strict contractivity. The assertion then follows by Banach's fixed-point theorem; cf., e.g., [9]}. Note that this implies uniqueness, first of all, only on \(M\) . Global uniqueness in \(X^{\textup {W}}\)  follows analogously to the standard uniqueness argument in the Picard-Lindelöf Theorem under a local Lipschitz condition; see, e.g., [9]}.
We emphasize that smallness of the initial data is only required in a weaker norm than the one of \(X^{\textup {W}}_0\) . The other smallness constraints in Theorem REF  can be achieved by making either \(\Vert (p_0,p_1)\Vert _{X^{\textup {W}}_0}\)  small or final time \(T\)  short enough. This theoretical framework thus agrees well with ultrasonic applications, where typically the final time will be short and the data smooth, but not necessarily small; see, for example, [6]} and [12]}.
When \(n=1\) , already the space \(H^1(\Omega )\)  embeds into \(L^\infty (\Omega )\) , which allows us to simplify the theoretical framework. Thus we expect that the results of Theorem REF  can be sharpened in that case; however, given the real-world ultrasonic applications that motivate our work, we do not pursue that path here.

The inviscid limit of the Westervelt equation
Equipped with a uniform bound in \(b\) , we are now ready to prove a limiting result for the Westervelt equation as \(b \rightarrow 0^+\) . To formulate the result, we introduce the space
\(\textup {E}=W^{1,\infty }(0,T;L^2(\Omega ))\cap L^\infty (0,T;H_0^1(\Omega ))\) 
equipped with the standard energy norm for the wave equation
\( \Vert p\Vert ^2_{\textup {E}}= \sup _{t\in (0,T)} \Vert p_t(t)\Vert _{L^2}^2+\sup _{t\in (0,T)}\Vert \nabla p(t)\Vert _{L^2}^2.\) 
Theorem 5.1 
Under the conditions of Theorem REF , the family of solutions \(\lbrace p^{(b)}\rbrace _{b>0}\)  to the Westervelt equation converges in the topology induced by the energy norm for the wave equation to a solution \(p\)  of the inviscid Westervelt equation at a linear rate
\(\Vert p^{(b)}-p\Vert _{\textup {E}}\lesssim b \mbox{ as } b\rightarrow 0.\) 

Following the general approach of [13]}, the statement follows by proving that \(\lbrace p^{(b)}\rbrace \)  is a Cauchy sequence in \(\textup {E}\) .
Let \(b, b^{\prime } \in (0, \overline{b})\) . Let \(p^{(b)}\)  and \(p^{(b^{\prime })}\)  be the solutions of the Westervelt equation with the sound diffusivity \(b\)  and \(b^{\prime }\) , respectively, and with homogeneous Dirichlet data. We assume the initial conditions and final time are chosen so that the assumptions of Theorem REF  are satisfied. Since these conditions are independent of \(b\) , solutions exist on a maximal common interval \([0,T]\)  with \(T>0\)  for all \(b, b^{\prime } \in [0, \overline{b})\) .
Then \(\overline{p}=p^{(b)}-p^{(b^{\prime })}\)  solves the equation
\( (1-k p^{(b)} )\overline{p}_{tt}-c^2 \Delta \overline{p}-b \Delta \overline{p}_t=k\overline{p}_t (p_t^{(b)}+p^{(b^{\prime })}_t)+k \overline{p}p^{(b^{\prime })} _{tt}+(b-b^{\prime })\Delta p_t^{(b^{\prime })}.\) 
Testing with \(\overline{p}_t\)  and integrating over space and time leads to
\(\begin{aligned}&\frac{1}{2}\Vert \sqrt{\alpha (t)}\overline{p}_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\Vert \nabla \overline{p}(t)\Vert _{L^2}^2+b \Vert \nabla \overline{p}_t\Vert _{L^2 (L^2)}^2 \\=&\,\begin{multlined}[t]-\frac{1}{2}k\int _0^t \int _{\Omega }p^{(b)}_t \overline{p}_t^2 \, \textup {d}x\textup {d}s+ k\int _0^t \int _{\Omega }(p_t^{(b)}+p_t^{(b^{\prime })})\overline{p}_t^2 \, \textup {d}x\textup {d}s+k\int _0^t \int _{\Omega }p _{tt}^{(b^{\prime })} \overline{p}\overline{p}_t \, \textup {d}x\textup {d}s\\+ (b-b^{\prime })\int _0^t \int _{\Omega }^{(b^{\prime })} \overline{p}_t \, \textup {d}x\textup {d}s. \end{multlined}\end{aligned}\) 
From here, by using Young's inequality and then taking the supremum over \((0,\tau )\) , we have the estimate
\( \begin{aligned}&\sup _{t\in (0,\tau )} \Vert \overline{p}_t(t)\Vert _{L^2}^2+\sup _{t\in (0,\tau )}\Vert \nabla \overline{p}(t)\Vert _{L^2}^2 +b \int _0^\tau \Vert \nabla \overline{p}_t\Vert _{L^2}^2\, \textup {d} s \\\lesssim &\,\begin{multlined}[t] (\Vert p^{(b)}_t\Vert _{L^\infty (L^\infty )}+\Vert p_t^{(b^{\prime })}\Vert _{L^\infty (L^\infty )} )\int _0^\tau \Vert \overline{p}_t\Vert _{L^2}^2 \, \textup {d} s \\+\varepsilon \sup _{s\in (0,\tau )} \Vert \overline{p}_t(s)\Vert _{L^2}^2+\Vert p _{tt}^{(b^{\prime })}\Vert _{L^2 (L^4)}^2 \int _0^\tau \Vert \overline{p}\Vert _{L^4}^2\, \textup {d} s \\+ (b-b^{\prime })^2 \Vert \Delta p_t^{(b^{\prime })}\Vert _{L^2 (L^2)}^2+ \int _0^\tau \Vert \overline{p}_t\Vert _{L^2}^2\, \textup {d} s , \end{multlined}\end{aligned}\) 
with \(\varepsilon >0\)  small enough, but independent of \(b\) .
Dominating the \(\varepsilon \)  term on the right-hand side by the first term on the left-hand side and using continuous embeddings as well as the uniform bounds \(\Vert p^{(b)}\Vert _{X^{\textup {W}}},\, \Vert p^{(b^{\prime })}\Vert _{X^{\textup {W}}}\, \lesssim R\)  resulting from Theorem REF , we conclude that
\( \begin{aligned}&\sup _{t\in (0,\tau )} \Vert \overline{p}_t(t)\Vert _{L^2}^2+\sup _{t\in (0,\tau )}\Vert \nabla \overline{p}(t)\Vert _{L^2}^2 +b \int _0^\tau \Vert \nabla \overline{p}_t\Vert _{L^2}^2\, \textup {d} s \\\lesssim &\,\begin{multlined}[t] R\int _0^\tau \Vert \overline{p}_t\Vert _{L^2}^2 \, \textup {d} s + R\int _0^\tau \Vert \nabla \overline{p}\Vert _{L^2}^2\, \textup {d} s + R^2(b-b^{\prime })^2 \end{multlined}\end{aligned}\) 
since \(p^{(b)}\) , \(p^{(b^{\prime })}\in M\) . Gronwall's inequality therefore yields
\(\Vert p^{(b)}-p^{(b^{\prime })}\Vert _{\textup {E}}=\Vert \overline{p}\Vert _{\textup {E}}\lesssim |b-b^{\prime }|.\) 
This estimate, which remains valid for \(b^{\prime }=0\)  also yields the convergence rate
\(\Vert p^{(b)}-p\Vert _{\textup {E}}=\Vert \overline{p}\Vert _{\textup {E}}\lesssim b,\) 
where \(p\)  is a solution to the inviscid version of the Westervelt equation.
Remark 2 (Convergence in a higher-order norm)  
Similarly to the proof of contractivity in Proposition REF , we can also establish convergence with respect to the topology used there
\(X=H^2(0,T;L^2(\Omega ))\cap W^{1,\infty }(0,T;H_0^1(\Omega ))\cap L^\infty (0,T;{H_\diamondsuit ^2(\Omega )});\) 
that is, a stronger topology than \(\textup {E}\) , with a slower convergence rate
\(\Vert p^{(b)}-p\Vert _{X}\lesssim \sqrt{b} \mbox{ as } b\rightarrow 0.\) 
Indeed, estimating the right-hand side terms up to the last one in (REF ) goes through analogously to the estimates for the difference equation (REF ); cf. (REF ). For the last term we obtain
\(\Vert (b-b^{\prime })\Delta p_t^{(b^{\prime })}\Vert _{L^2 (L^2)} \le |b-b^{\prime }|\, b^{-1/2}R,\) 
with \(R\)  as in (REF ); see also (REF ) and (REF ). As in the proof of Proposition REF , this yields
\(\Vert p^{(b)}-p\Vert _{X}\lesssim |b-b^{\prime }| b^{-1/2}.\) 
Thus, with \(b^{\prime }=0\)  above, we obtain the convergence rate (REF ).


Analysis of the Kuznetsov equation
We wish to extend our considerations of the vanishing sound diffusivity dynamics to the Kuznetsov equation next. As a by-product, we will also obtain results for the Westervelt equation in the acoustic potential formulation (REF ) by setting \(\sigma =0\)  and \(\kappa =\frac{2}{c^{2}}(\frac{B}{2A}+1)\)  in (REF ).
Compared to the analysis of the Westervelt equation, now the factor next to the second time derivative involves \(\psi _t\) . This implies that a uniform bound for \(\Vert \psi _t\Vert _{L^\infty (L^\infty )}\)  is needed. Together with the presence of quadratic gradient nonlinearity, higher-order energy analysis is required in this case to arrive in the end at a uniform bound with respect to the parameter \(b\) .
Energy analysis of the linearized Kuznetsov equation
Similarly to before, we begin by studying a linear non-degenerate version of the nonlinear equation in question. We consider the following linearization:
\( \left\lbrace \begin{aligned}\alpha \psi _{tt} -c^2- b _t=&\, \sigma \nabla \phi \cdot \nabla \psi _t &\quad \text{ in }\Omega \times (0,T), \\[1mm]\psi =&\,0 &\quad \text{ on } \partial \Omega \times (0,T),\\[1mm](\psi , \psi _t)=&\,(\psi _0, \psi _1) &\quad \mbox{ in }\Omega \times \lbrace 0\rbrace ,\end{aligned} \right.\) 
with \(\alpha =1-\kappa \phi _t\)  and \(\kappa \in \mathbb {R}\) . By abbreviating \(p=\psi _t\) , \(q=\phi _t\) , the PDE above can be rewritten as
\(\alpha p_t - b p̥ -c^2= \sigma \nabla \phi \cdot \nabla p\) 
with \(\alpha =1-\kappa q \) .
The functions \(\phi \)  and \(q=\phi _t\)  are required to have the following regularity:
\(\phi \in L^\infty (0,T; {H_\diamondsuit ^4(\Omega )}), \ q\in L^\infty (0,T;{H_\diamondsuit ^3(\Omega )}), \ q_t\in L^\infty (0,T;{H_\diamondsuit ^2(\Omega )}),\) 
which implies that the coefficient \(\alpha \)  has the regularity
\(\begin{aligned}&\alpha \in X^{\textup {K}}_{\alpha }=L^\infty (0,T;{H_\diamondsuit ^3(\Omega )})\cap W^{1,\infty }(0,T;{H_\diamondsuit ^2(\Omega )});\end{aligned}\) 
cf. (REF ).
Again, a compatibility condition on the initial data will be needed, which here reads as
\(\psi _{tt}(0)=\psi _2 =\alpha (0)^{-1} (b _1 +c^2_0 + \sigma \nabla \phi (0)\cdot \nabla \psi _1)\,.\) 
We are now ready to state a well-posedness result for this linear problem.
Proposition 6.1 
Assume that the domain \(\Omega \subset \mathbb {R}^n\) , where \(n \in \lbrace 1, 2, 3\rbrace \) , is bounded and \(C^4\)  regular. Let \(b \in [0, \bar{b})\)  and \(T>0\) . Let \(\phi \in L^\infty (0,T;{H_\diamondsuit ^4(\Omega )})\) . Assume that \(\alpha \in X^{\textup {K}}_{\alpha }\)  and that there exist \(\underline{\alpha }\) , \(\overline{\alpha }>0\)  such that
\( \underline{\alpha }\le \alpha (x, t)\le \overline{\alpha }\ \mbox{ on }\Omega \ \ \text{ a.e. in } \Omega \times (0,T).\) 
Furthermore, assume that
\( (\psi _0, \psi _1) \in X^{\textup {K}}_0={H_\diamondsuit ^4(\Omega )}\times {H_\diamondsuit ^3(\Omega )}\) 
and (REF ) holds.
Then there exists a unique solution \(\psi \)  of the problem (REF ) such that
\(\begin{aligned}\psi \in \, X^{\textup {K}}=& \,\begin{multlined}[t] H^3(0,T;H_0^1(\Omega ))\cap W^{2,\infty }(0,T;{H_\diamondsuit ^2(\Omega )})\\\cap W^{1,\infty }(0,T;{H_\diamondsuit ^3(\Omega )})\cap L^\infty (0,T;{H_\diamondsuit ^4(\Omega )}).\end{multlined}\end{aligned}\) 
Furthermore, this solution satisfies the estimate
\( \begin{aligned}& \begin{multlined}[t] \Vert \nabla \psi _{ttt}\Vert _{L^2 (L^2)}^2\,+\operatornamewithlimits{ess\,sup}_{t \in (0,T)} \Vert _{tt} (t)\Vert _{L^2}^2\,+\operatornamewithlimits{ess\,sup}_{t \in (0,T)} \Vert \nabla _t(t)\Vert _{L^2}^2 \\+\operatornamewithlimits{ess\,sup}_{t \in (0,T)} \Vert 2 \psi (t)\Vert _{L^2}^2+b\Vert \nabla _{tt}\Vert _{L^2 (L^2)}^2 +b\Vert 2 \psi _t\Vert _{L^2 (L^2)}^2\end{multlined}\\\lesssim &\, \begin{multlined}[t] \Vert 2 \psi _0\Vert _{L^2}^2+\Vert \nabla _1\Vert _{L^2}^2+\Vert \nabla \phi (0)\cdot \nabla \psi _1]\Vert _{L^2}^2,\end{multlined}\end{aligned}\) 
where
\(\Vert \nabla \phi (0)\cdot \nabla \psi _1]\Vert _{L^2}\le 2(C_{H^2, L^\infty }+C_{H^1, L^4}^2)\Vert \nabla (0)\Vert _{L^2}\Vert \nabla _1\Vert _{L^2}.\) 
The hidden constant in (REF ) tends to infinity as \(T \rightarrow \infty \) , but does not depend on the parameter \(b\) .

We start by proving existence of solutions to the time-differentiated version of our original problem (REF ). We thus consider
\( \left\lbrace \begin{aligned}\alpha p_{tt} - b p̥_t -c^2p̥ =&-\alpha _t p_t + \sigma \nabla q\cdot \nabla p+\sigma \nabla \phi \cdot \nabla p_t \hspace*{-11.38109pt} &&\ \text{ in }\Omega \times (0,T), \\[1mm]p=&\,0 &&\ \text{ on } \partial \Omega \times (0,T),\\[1mm](p, p_t)=&\,(\psi _1, \psi _2) &&\ \mbox{ in }\Omega \times \lbrace 0\rbrace ,\end{aligned} \right.\) 
where the function \(\psi _2\)  is defined by (REF ).
The proof can be carried out as before via smooth Faedo–Galerkin approximations in space, by projecting the problem (REF ) onto the span \(V_n\)  of the first \(n\)  eigenfunctions of the Laplacian pointwise in time.
We focus here on deriving the crucial uniform bound for the approximate solution since the remaining steps are analogous to the proof of Proposition REF . For simplicity of notation, we drop the superscript \(n\)  in the approximate solution.
In the first step, we multiply (REF ) by \(2 p_t\)  and integrate over \(\Omega \) :
\((\alpha p_{tt} - b p̥_t -c^2p̥, 2 p_t)_{L^2}=(-\alpha _t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t, 2 p_t)_{L^2}.\) 
When deriving the energy bound, we can rely on the identity
\(\begin{aligned}(\alpha p_{tt}, 2 p_t)_{L^2}=&\,\, (\alpha p_{tt}], p̥_t)_{L^2} \\=&\, \frac{\textup {d}}{\textup {d}t}\frac{1}{2} (\alpha p̥_t, p̥_t)_{L^2} +\frac{1}{2} \kappa (q_t p̥_t, p̥_t)_{L^2}-(\kappa q̥\, p_{tt} +\kappa \nabla q\cdot \nabla p_{tt}, p̥_t)_{L^2}\\=&\, \frac{\textup {d}}{\textup {d}t}\frac{1}{2} (\alpha p̥_t, p̥_t)_{L^2} + (f_1(\phi ,q,p), p̥_t)_{L^2},\end{aligned}\) 
where we have used the fact that \(p̥\)  vanishes on the boundary for smooth approximations of the problem and introduced the short-hand notation
\(\begin{aligned}f_1(\phi ,q,p) = \frac{1}{2} \kappa q_t -\kappa q̥ p_{tt}- \kappa \nabla q \cdot \nabla p_{tt}.\end{aligned}\) 
Recalling that \(p_{tt}==p̥=0\)  on \(\partial \Omega \) , which via the PDE implies that also \(\nabla q \cdot \nabla p+\nabla \phi \cdot \nabla p_t\)  vanishes on the boundary, we further find that
\(\begin{aligned}&(-\alpha _t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t, 2 p_t)_{L^2}\\=&\,\begin{multlined}[t](-\alpha _t p_t], )_{L^2}+\sigma (\nabla q\cdot \nabla p + \nabla \phi \cdot \nabla p_t], )_{L^2}.\end{multlined}\end{aligned}\) 
Furthermore, we have
\(\begin{aligned}&-\alpha _t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t]\\=&\,\begin{multlined}[t] \kappa \Delta q_t \, p_t + 2\kappa \nabla q_t\cdot \nabla p_t + \kappa q_t\,p̥_t+ \sigma \nabla q̥\cdot \nabla p+ 2\sigma D^2 q:D^2 p\\+ \sigma \nabla q\cdot \nabla p̥+ \sigma \nabla \cdot \nabla p_t+ 2\sigma D^2 \phi :D^2 p_t+ \sigma \nabla \phi \cdot \nabla p̥_t \end{multlined}\\=&\, f_2(\phi ,q,p)+\sigma \nabla \phi \cdot \nabla p̥_t,\end{aligned}\) 
where we have introduced another short-hand notation
\(\begin{aligned}f_2(\phi ,q,p)= \kappa \Delta q_t \, p_t + 2\kappa \nabla q_t\cdot \nabla p_t + \kappa q_t\,p̥_t+ \sigma \nabla q̥\cdot \nabla p+ 2\sigma D^2 q:D^2 p\\+ \sigma \nabla q\cdot \nabla p̥+ \sigma \nabla \cdot \nabla p_t+ 2\sigma D^2 \phi :D^2 p_t\,.\end{aligned}\) 
By elliptic regularity the Hessian \(D^2 v=(\partial _{x_i} \partial _{x_j} v)_{i,j}\)  satisfies
\(\Vert D^2 v\Vert _{L^p}\le C_{\textup {H}} \Vert \Delta v\Vert _{L^p}\quad \mbox{ for all }v\in H_0^1(\Omega )\cap W^{2,p}(\Omega )\, \quad p\in (1,6],\) 
see, e.g., [14]}.

Due to the PDE, we know that
\(\begin{aligned}p_{tt}=&\,\alpha ^{-1}\left(b p̥_t +c^2p̥ + \kappa q_t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t\right),\\\nabla p_{tt}=&\, \begin{multlined}[t]\kappa \alpha ^{-2}\nabla q\,\left(b p̥_t +c^2p̥ + \kappa q_t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t\right)\\+\alpha ^{-1}\left(b \nabla p̥_t +c^2\nabla p̥ + \kappa \nabla q_t p_t+ \kappa q_t \nabla p_t+ \sigma D^2 q\,\nabla p \right. \\ \left.+ \sigma D^2 p\,\nabla q + \sigma D^2\phi \,\nabla p_t + \sigma D^2 p_t\,\nabla \phi \right). \end{multlined}\end{aligned}\) 
Altogether, we have the energy identity
\(\begin{aligned}&\frac{1}{2}\frac{\textup {d}}{\textup {d}t}\Vert \sqrt{\alpha }p̥_t\Vert _{L^2}^2 + b \Vert \nabla \Delta p_t\Vert _{L^2}^2 + c^2 \frac{1}{2}\frac{\textup {d}}{\textup {d}t} \Vert \nabla p̥\Vert _{L^2}^2 \\[1mm]&= \begin{multlined}[t](-f_1(\phi ,q,p)+f_2(\phi ,q,p)+\sigma \nabla \phi \cdot \nabla p̥_t, p̥_t)_{L^2}.\end{multlined}\end{aligned}\) 
We can estimate the right-hand side by employing the following bound:
\(\begin{aligned}\Vert f_1(\phi ,q,p)\Vert _{L^2}\le &\, \begin{multlined}[t] \frac{1}{2}|\kappa | C_{H^2, L^\infty }\Vert q̥_t\Vert _{L^2}\Vert p̥_t\Vert _{L^2}\\+ (|{\kappa }|C_{H^2, L^\infty }+|\kappa |C_{H^1, L^4}^2) \Vert \nabla q̥\Vert _{L^2} \Vert \nabla p_{tt}\Vert _{L^2}. \end{multlined}\end{aligned}\) 
Furthermore, by virtue of identities (REF ), we have
\(\begin{aligned}\Vert \nabla p_{tt}\Vert _{L^2}\le &\, \begin{multlined}[t]\frac{|\kappa |}{\underline{\alpha }^2} C_{H^2, L^\infty }\Vert \nabla q̥\Vert _{L^2}\left(b\Vert p̥_t\Vert _{L^2}+c^2\Vert p̥\Vert _{L^2}+C_{H^1, L^4}^2 \left(|\kappa |\Vert \nabla q_t\Vert _{L^2}\Vert \nabla p_t\Vert _{L^2} \right. \right.\\ \left. \left.+ |\sigma |\Vert q̥\Vert _{L^2}\Vert p̥\Vert _{L^2} + |\sigma |\Vert \Vert _{L^2}\Vert p̥_t\Vert _{L^2}\right) \vphantom{C_{H^2, L^\infty }}\right)\\+\frac{1}{\underline{\alpha }} \left(b \Vert \nabla p̥_t\Vert _{L^2}+c^2\Vert \nabla p̥\Vert _{L^2}+2|\kappa | C_{H^1, L^4}C_{\textup {PF}}^{1/2} \Vert q̥_t\Vert _{L^2}\Vert p̥_t\Vert _{L^2}\right. \\ \left.+|\sigma |C_{\textup {H}} (C_{H^2, L^\infty }+C_{H^1, L^4}^2) (\Vert \nabla q̥\Vert _{L^2}\Vert p̥\Vert _{L^2}+\Vert \nabla \Vert _{L^2}\Vert p̥_t\Vert _{L^2})\right). \end{multlined}\end{aligned}\) 
We note that also
\(\begin{aligned}\Vert f_2(\phi ,q,p)\Vert _{L^2}\le &\, \begin{multlined}[t]2|\kappa |(C_{H^2, L^\infty }+C_{H^1, L^4}^2) \Vert q̥_t\Vert _{L^2}\Vert p̥_t\Vert _{L^2} \\+ 2|\sigma |(C_{H^2, L^\infty }+C_{\textup {H}}^2 C_{H^1, L^4}^2) \Vert \nabla q̥\Vert _{L^2}\Vert \nabla p̥\Vert _{L^2} \\+ |\sigma |(C_{\textup {H}} C_{H^2, L^\infty }+2 C_{H^1, L^4}^2) \Vert 2\phi \Vert _{L^2}\Vert p̥_t\Vert _{L^2} \end{multlined}\end{aligned}\) 
and
\(\begin{aligned}2 (\nabla \phi \cdot \nabla p̥_t, p̥_t)_{L^2} = (\nabla \phi , \nabla (p̥_t)^2)_{L^2} =&\, (, (p̥_t)^2)_{L^2}\\\le &\, C_{H^2, L^\infty }\Vert \Delta ^2\phi \Vert _{L^2}\Vert p̥_t\Vert _{L^2}^2.\end{aligned}\) 
Under the regularity assumptions (REF ), similarly to the proof of Proposition REF , by using Young's and Gronwall's inequalities, we arrive at an energy estimate of the form
\( \begin{aligned}& \begin{multlined}[t] \Vert \nabla p_{tt}\Vert _{L^2 (L^2)}^2\,+\sup _{t \in (0,T)} \Vert (t)\Vert _{L^2}^2\,+\sup _{t \in (0,T)} \Vert \nabla p̥(t)\Vert _{L^2}^2+b\Vert \nabla \Vert _{L^2 (L^2)}^2\end{multlined}\\\lesssim &\, \begin{multlined}[t] \Vert \nabla _1\Vert _{L^2}^2+\Vert _2\Vert _{L^2}^2,\end{multlined}\end{aligned}\) 
first of all, for the Galerkin approximations. Via weak limits we obtain existence, uniqueness, and the energy estimate (REF ) also for a solution to (REF ).
Integrating with respect to time and using compatibility of the initial data (REF ) yields a unique solution to (REF ).
Additional regularity
The terms containing \(\Vert \Delta ^2\phi \Vert _{L^2}\Vert p̥_t\Vert _{L^2}\)  above and in \(f_2\)  show that another energy estimate is needed to obtain an overall bound that is independent of \(b\)  for the nonlinear problem. To this end, we exploit elliptic regularity and the fact that by the PDE, \(\)  satisfies
\((b\partial _t+c^2)= r \mbox{ with } r=\alpha p_t -\sigma \nabla \phi \cdot \nabla p.\) 
Thus in case \(b>0\) , we have
\((t) = g_0(t) (0) + \int _0^t g^{\prime }(t-s) r(s)\, \textup {d} s \) 
with
\(g(t)=-c^{-2} \exp {\left(-\frac{c^2}{b} t\right)},\qquad g_0(t)=\exp {\left(-\frac{c^2}{b} t\right)}\in [0,1].\) 
By positivity of \(g^{\prime }\) , we have
\(\Vert g^{\prime }\Vert _{L^1(0,t)}=\int _0^t g^{\prime }(s)\, \textup {d} s = g(t)-g(0) = c^{-2}\left(1-\exp {\left(-\frac{c^2}{b} t\right)}\right)\le c^2.\) 
Therefore,
\(\Vert 2 \psi (t)\Vert _{L^2}\le \Vert 2 \psi (0)\Vert _{L^2} + c^{-2} \Vert r̥\Vert _{L^\infty (0,t;L^2(\Omega )}.\) 
In case \(b=0\) , estimate (REF ) (without the \(\Vert 2 \psi (0)\Vert _{L^2}\)  term) immediately follows from (REF ). The right hand-side norm can be estimated as follows:
\(\begin{aligned}&\Vert r̥\Vert _{L^\infty (L^2)}\\=&\, \Vert \kappa q̥ \, p_t + 2\kappa \nabla q\cdot \nabla p_t + \alpha \,p̥_t+ \sigma \nabla \cdot \nabla p+ 2\sigma D^2 \phi :D^2 p+ \sigma \nabla \phi \cdot \nabla p̥\Vert _{L^\infty (L^2)}\\\le &\, \begin{multlined}[t]{(1+|\kappa | (2C_{H^2, L^\infty }+2C_{\textup {H}}^2 C_{H^1, L^4}^2)}\Vert q̥\Vert _{L^\infty (L^2)} )\Vert p̥_t\Vert _{L^\infty (L^2)} \\+2 |\sigma | ({C_{\textup {H}}}C_{H^2, L^\infty }+C_{\textup {H}}^2 C_{H^1, L^4}^2)\Vert \nabla \Vert _{L^\infty (L^2)} \Vert \nabla p̥\Vert _{L^\infty (L^2)}\,,\end{multlined}\end{aligned}\) 
and therefore bounded by means of the already established estimate (REF ).

Here we have used the elliptic regularity result (see, e.g., [15]})
\(\Vert v\Vert _{H^3}\le C_{\textup {H}} \Vert (-\Delta )^{3/2} v\Vert _{L^2}\quad \mbox{ for all }v\in H_0^1(\Omega )\cap H^3(\Omega ),\) 
and estimated
\(\Vert \nabla p\Vert _{L^\infty }\le C_{H^2, L^\infty } \Vert p\Vert _{H^3}\le C_{H^2, L^\infty } C_{\textup {H}} \Vert (-\Delta )^{3/2} p\Vert _{L^2}= C_{H^2, L^\infty } C_{\textup {H}} \Vert \nabla \Delta p\Vert _{L^2}\,,\) 
and in the same manner bounded \(\Vert \nabla \phi \Vert _{L^\infty }\) , due to the fact that \(\Delta p\)  and \(\Delta \phi \)  vanish on \(\partial \Omega \) .

Furthermore, by (REF ), the function \(\psi \)  satisfies the boundary conditions
\(\begin{aligned}&\psi \vert _{\partial \Omega }=0\,, \\&\vert _{\partial \Omega }={\mathrm {bdy}}:={\left\lbrace \begin{array}{ll}g_0(t) \vert _{\partial \Omega }(0) + \int _0^t g^{\prime }(t-s)(\sigma \nabla \phi \cdot \nabla p)\vert _{\partial \Omega }(s)\, \textup {d} s &\mbox{ if }b>0\\(\sigma \nabla \phi \cdot \nabla p)\vert _{\partial \Omega }&\mbox{ if }b=0\,,\end{array}\right.}\end{aligned}\) 
where
\(\begin{aligned}\Vert (\sigma \nabla \phi \cdot \nabla p)\vert _{\partial \Omega }\Vert _{L^\infty (0,T;H^{3/2}(\partial \Omega ))}\le C_{\textup {tr}} \Vert \sigma \nabla \phi \cdot \nabla p\Vert _{L^\infty (0,T;{H^2(\Omega )})}\end{aligned}\) 
can again be estimated by means of (REF ).
Thus we can invoke higher elliptic regularity [16]} to first conclude
\(\Vert \psi \Vert _{L^\infty (0,T;H^4(\Omega ))}\le C_{\textup {H}}\Bigl ( \Vert \Vert _{L^\infty (0,T;H^2(\Omega )}+\Vert \psi \Vert _{L^\infty (L^2)}\Bigr )\) 
and then further estimate
\(\Vert \Vert _{L^\infty (0,T;H^2(\Omega ))}\le C_{\textup {H}}\Bigl ( \Vert r̥\Vert _{L^\infty (L^2)} + {\Vert \mathrm {bdy}\Vert _{L^\infty (H^{3/2}(\partial \Omega ))}} + \Vert \psi \Vert _{L^\infty (L^2)}\Bigr ).\) 
Altogether, by taking into account the derived bounds, we obtain the energy estimate (REF ); cf. (REF ), where we estimate \(\Vert _2\Vert _{L^2}^2\)  by means of (REF ).

Uniform bounds for the Kuznetsov equation
We next employ a fixed-point argument to analyze the Kuznetsov equation for small data as well as obtain a uniform in \(b\)  energy bound for its solution. To this end, we introduce the mapping
\(\mathcal {T}_{\textup {K}}:\phi \mapsto \psi ,\) 
where, similarly to before, \(\phi \)  will belong to a suitably chosen ball in the space \(X^{\textup {K}}\) , and \(\psi \)  will solve the linearized Kuznetsov equation
\( \alpha \psi _{tt} -c^2- b _t =\, \sigma \nabla \phi \cdot \nabla \psi _t\) 
with \(\alpha =1-\kappa \phi _t\)  and initial data \((\psi (0), \psi _t(0))=(\phi (0), \phi _t(0))=(\psi _0, \psi _1)\) . We note that the mapping \(\mathcal {T}_{\textup {K}}\)  is well-defined on account of Proposition REF , whose assumptions we verify below.
Theorem 6.1 
Let \(\Omega \subset \mathbb {R}^n\) , where \(n \in \lbrace 1, 2, 3\rbrace \) , be bounded and \(C^4\)  regular. Let \(\kappa \) , \(\sigma \in \mathbb {R}\) . Furthermore, let \(b \in [0, \bar{b})\)  and let the initial data \((\psi _0, \psi _1) \)  satisfy the regularity assumption (REF ) with
\(\Vert \psi _0\Vert ^2_{H^4}+\Vert \psi _1\Vert ^2_{H^3} \le \delta ^2.\) 
There exist \(\tilde{\delta }>0\)  and a final time \(T=T(\delta )>0\) , independent of \(b\) , such that if
\({\Vert \psi _0\Vert ^2_{H^2}+\Vert \psi _1\Vert ^2_{H^2} \le \tilde{\delta }^2,}\) 
then there exists a unique solution \(\psi \in X^{\textup {K}}\)  of the problem
\( \left\lbrace \begin{aligned}\psi _{tt}- b _t - c^2=&\, \frac{1}{2} (\kappa \psi _t^2 + \sigma |\nabla \psi |^2)_{t} &&\quad \text{ in }\Omega \times (0,T), \\[1mm]\psi =&\,0 &&\quad \text{ on } \partial \Omega \times (0,T),\\[1mm](\psi , \psi _t)=&\,(\psi _0, \psi _1) &&\quad \mbox{ in }\Omega \times \lbrace 0\rbrace ,\end{aligned} \right.\) 
where \(X^{\textup {K}}\)  is defined in (REF ). Furthermore, the solution \(\psi \)  satisfies the estimate (REF ).

We begin by proving that the mapping \(\mathcal {T}_{\textup {K}}:\phi \mapsto \psi \)  is well-defined and a self-mapping on
\(\begin{aligned}M_{\textup {K}} = \left\lbrace  \vphantom{\sup _{t \in (0,T)} \Vert 2 \psi (t)\Vert _{L^2}^2\le R_{\textup {K}}^2 }\right. \phi \in X^{\textup {K}}\, : \, &\phi (0)= \psi _0, \ \phi _t(0)=\psi _1,\\&\ \left. \vphantom{X^{\textup {K}}}\Vert \phi _t\Vert _{L^\infty (L^\infty )} \le \frac{1}{2 |\kappa |}, \right.\ \left.\Vert \phi \Vert _{X^{\textup {K}}} \le R_{\textup {K}}\ \vphantom{\sup _{t \in (0,T)} \Vert 2 \psi (t)\Vert _{L^2}^2\le R_{\textup {K}}^2 } \right\rbrace ,\end{aligned}\) 
where the radius \(R_{\textup {K}}\)  will be specified below. We first note that the conditions of Proposition REF  are satisfied. Indeed, regularity assumptions (REF ) and (REF ) hold with the uniform bounds
\(\begin{aligned}\Vert \phi \Vert _{L^\infty (H_\diamondsuit ^4)} \le \, R_{\textup {K}}, \qquad \Vert \alpha \Vert _{X^{\textup {K}}_{\alpha }} \lesssim \, 1+ R_{\textup {K}}.\end{aligned}\) 
Furthermore, we have
\(\begin{aligned}\Vert 1-\alpha \Vert _{L^\infty (L^\infty )}\le |\kappa |\Vert \phi _t\Vert _{L^\infty (L^\infty )} \le \frac{1}{2}.\end{aligned}\) 
We can thus take \(\underline{\alpha }=\frac{1}{2}\)  and \(\overline{\alpha }=\frac{3}{2}\)  in (REF ), and so the non-degeneracy condition is fulfilled as well. Then Proposition REF  applied to \(\psi =\mathcal {T}_{\textup {K}}(\phi )\)  implies that the mapping is well-defined and, moreover,
\( \begin{aligned}\Vert \psi \Vert ^2_{X^{\textup {K}}} \le \, C_{\textup {lin}, \textup {K}}(T, R_{\textup {K}})(\Vert 2 \psi _0\Vert _{L^2}^2+\Vert \nabla _1\Vert _{L^2}^2+\Vert \psi _0\Vert _{L^2}^2),\end{aligned}\) 
where \(C_{\textup {lin}, \textup {K}}(T, R_{\textup {K}})\)  is determined by the hidden constant in the linear bound (REF )

and by inspection of (REF ) can be seen to have a form similar to the Westervelt case (REF )
\(C_{\textup {lin}}(T,R)= C_1 \exp {\left(C_2(|\kappa |+|\sigma |)(R_{\textup {K}}+R_{\textup {K}}^2+R_{\textup {K}}^3)T\right)},\) 
with constants \(C_1\)  and \(C_2\)  independent of \(R_{\textup {K}}\) , \(T\) , and \(b \in [0, \bar{b})\)  for fixed \(\bar{b}\) .

Therefore, provided that \(R_{\textup {K}}\)  and final time \(T\)  are chosen so that
\(C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}}) (\Vert 2 \psi _0\Vert _{L^2}^2+\Vert \nabla _1\Vert _{L^2}^2+\Vert \psi _0\Vert _{L^2}^2) \le R_{\textup {K}}^2,\) 
the solution \(\psi \)  satisfies the \(R_{\textup {K}}^2\)  bound in (REF ).
To obtain the \(1/{(2|\kappa |)}\)  bound, we will, similarly to the Westervelt case, derive a uniform in \(b\)  estimate of \(\Vert \psi _t \Vert _{L^\infty (H^1)}\)  and then rely on Agmon's interpolation inequality. By multiplying the time-differentiated equation (REF ) with \(p_{t}\)  and integrating over space and time, we arive at
\( \begin{aligned}&\frac{1}{2} \left\lbrace \Vert \sqrt{\alpha (t)}p_t(t)\Vert ^2_{L^2}+c^2\Vert \nabla p(t)\Vert ^2_{L^2} \right\rbrace \Big \vert _0^t+b \int _0^t\Vert \nabla p_t\Vert ^2\, \textup {d}x\textup {d}s\\=&-\frac{1}{2} \int _0^t(\alpha _t p_t, p_t)_{L^2}\, \textup {d} s +\sigma \int _0^t (\nabla q \cdot \nabla p+\nabla \phi \cdot \nabla p_t, p_t)\, \textup {d} s \\\lesssim &\,\begin{multlined}[t] \Vert \alpha _t\Vert _{L^\infty (L^\infty )}\Vert p_t\Vert ^2_{L^2(0,t;L^2)}+\Vert \nabla q\Vert _{L^\infty (L^\infty )}\Vert \nabla p\Vert _{L^2(0, t;L^2)}\Vert p_t\Vert _{L^2(0, t;L^2)}\\+\sigma \int _0^t (\nabla \phi \cdot \nabla p_t, p_t)\, \textup {d} s . \end{multlined}\end{aligned}\) 
To estimate the last term on the right, we can first rely on the identity
\( \begin{aligned}& \int _0^t \int _{\Omega } (\nabla \phi \cdot \nabla p_t) p_t \, \textup {d}x\textup {d}s\\=&\, \int _0^t \int _{\Omega } (- ) p_t p_t \, \textup {d}x\textup {d}s- \int _0^t \int _{\Omega } p_t (\nabla \phi \cdot \nabla p_t) \, \textup {d}x\textup {d}s\end{aligned}\) 
since \(p_t=0\)  on \(\partial \Omega \) , which then leads to the bound
\( \begin{aligned}\left|2\int _0^t \int _{\Omega } (\nabla \phi \cdot \nabla \overline{\psi }_t) p_t \, \textup {d}x\textup {d}s\right|=&\, \left| \int _0^t \int _{\Omega } (- ) \overline{\psi }_t p_t \, \textup {d}x\textup {d}s\right|\\\le &\, \Vert \Vert _{L^\infty (0,t; L^\infty )}\Vert p_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
Thus, utilizing the above estimate in (REF ) and employing Gronwall's inequality leads to
\(\begin{aligned}&\Vert p_t(t)\Vert ^2_{L^2}+\Vert \nabla p(t)\Vert ^2_{L^2} \\\le &\, \tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_1\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2 R_{\textup {K}}T)(\Vert \psi _2\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} )\\\le &\,\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_1\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2 R_{\textup {K}}T)(\Vert \alpha (0)^{-1} (b _1 +c^2_0 + \sigma \nabla \psi _0\cdot \nabla \psi _1)\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} ),\end{aligned}\) 
where we have also relied on the compatibility of initial data (REF ) in the last line. From here, noting that
\(\Vert \nabla \psi _0\cdot \nabla \psi _1\Vert _{L^2} \le \Vert \nabla \psi _0\Vert _{L^4}\Vert \nabla \psi _1\Vert _{L^4} \lesssim \Vert \nabla \psi _0\Vert _{H^1}\Vert \nabla \psi _1\Vert _{H^1},\) 
we conclude that there exist positive constants \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2\)  and \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_3\) , independent of \(b\) , such that
\(\begin{aligned}\Vert p_t(t)\Vert ^2_{L^2}+\Vert \nabla p(t)\Vert ^2_{L^2}\le \,\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_3\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2 R_{\textup {K}}T)(\Vert _1\Vert ^2_{L^2} +\Vert _0\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} )\end{aligned}\) 
for all \(t \in [0,T]\) . The non-degeneracy bound follows from the above estimate and Agmon's inequality:
\(\Vert p\Vert _{L^\infty (L^\infty )} \le C_{\textup {A}}\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_4\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_5 RT)(\Vert _1\Vert ^2_{L^2} +\Vert _0\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} )^{1/4}R_{\textup {K}}^{1/2},\) 
for some positive constants \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_4\) , \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_5\)  independent of \(b\) , if we choose \(\tilde{\delta }>0\)  small enough, so that
\(C_{\textup {A}}\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_4\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_5 RT)\sqrt{\tilde{\delta }}\,R_{\textup {K}}^{1/2} \le \frac{1}{2 |\kappa |}.\) 
Therefore,
\(\mathcal {T}_{\textup {K}}(M_{\textup {K}}) \subset M_{\textup {K}}.\) 
We next prove strict contractivity of the mapping \(\mathcal {T}_{\textup {K}}\)  in the energy norm for sufficiently small data. To this end, we take \(\phi ^{(1)}\)  and \(\phi ^{(2)}\)  in \(M_{\textup {K}}\) , and denote their difference by \(\overline{\phi }= \phi ^{(1)} -\phi ^{(2)} \) . Let then \(\psi ^{(1)}=\mathcal {T}_{\textup {K}}(\phi ^{(1)})\)  and \(\psi ^{(2)}=\mathcal {T}_{\textup {K}}(\phi ^{(2)})\) . The difference \(\overline{\psi }=\psi ^{(1)} -\psi ^{(2)} \in X^{\textup {K}}\)  solves the equation
\( \begin{aligned}(1-\kappa \phi _t^{(1)} )\overline{\psi }_{tt}-c^2 \Delta \overline{\psi }-b \Delta \overline{\psi }_t= \kappa \overline{\phi }_t \psi ^{(2)}_{tt}+\sigma \nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t+\sigma \nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t\end{aligned}\) 
with zero initial data. We test this equation with \(\overline{\psi }_t\)  and integrate over space and \((0,t)\)  to arrive at the identity
\( \begin{aligned}&\frac{1}{2}\Vert \sqrt{\alpha (t)}\,\overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert _{L^2(0, t; L^2)}^2 \\=&\,\begin{multlined}[t] -\frac{\kappa }{2} \int _0^t \int _{\Omega }\phi _{tt}^{(1)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\overline{\phi }_t\psi _{tt}^{(2)} \overline{\psi }_t \, \textup {d}x\textup {d}s\\+ \sigma \int _0^t \int _{\Omega } (\nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t+\nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s, \end{multlined}\end{aligned}\) 
where \(\alpha (t)=1-\kappa \phi ^{(1)}_t(t)\) . To treat the \(\nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t\)  term on the right, we employ the following estimate:
\(\begin{aligned}\left|2\int _0^t \int _{\Omega } \nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right| \le &\, 2\Vert \nabla \psi ^{(1)}_t\Vert _{L^2(0,t ;L^\infty )} \Vert \nabla \overline{\phi }\Vert _{L^\infty (0,t; L^2)} \Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}.\end{aligned}\) 
Similarly to (REF )–(REF ), to estimate the \(\nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t\)  term, we can rely on
\( \begin{aligned}\left|2\int _0^t \int _{\Omega } (\nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s\right|=&\, \left| \int _0^t \int _{\Omega } (- ^{(2)}) \overline{\psi }_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right|\\\le &\, \Vert ^{(2)}\Vert _{L^\infty (0,t; L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
Additionally, we can bound the two \(\kappa \)  terms on the right-hand side of (REF ) as follows:
\(\begin{aligned}&\left|-\frac{1}{2}\kappa \int _0^t \int _{\Omega }\phi _{tt}^{(1)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(2)}\overline{\phi }_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right| \\\le &\,\begin{multlined}[t] \frac{1}{2} |\kappa | \Vert \phi _{tt}^{(1)}\Vert _{L^\infty (0,t; L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2 + |\kappa |\Vert \psi _{tt}^{(2)}\Vert _{L^2(0,t; L^\infty )} \Vert \overline{\phi }_t\Vert _{L^\infty (0,t; L^2)}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}. \end{multlined}\end{aligned}\) 
Employing the above estimates in (REF ), using Young's inequality, and then taking the supremum over
\(t \in (0, \tau )\)  leads to
\( \begin{aligned}&\frac{1}{2} \underline{\alpha }\sup _{t\in (0,\tau )}\Vert \overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\sup _{t\in (0,\tau )}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert ^2_{L^2(0,\tau ;L^2)} \\\le &\,\begin{multlined}[t] \frac{1}{2} |\kappa | \Vert \phi _{tt}^{(1)}\Vert _{L^\infty (L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,\tau ;L^2)}^2+\frac{1}{2} |\kappa \Vert \psi _{tt}^{(2)}\Vert _{L^2(L^\infty )}|\Vert \overline{\psi }_t\Vert _{L^2(0,\tau ;L^2)}^2 \\+ \frac{1}{2}|\kappa | \Vert \psi _{tt}^{(2)}\Vert _{L^2(L^\infty )}\Vert \overline{\phi }_t\Vert _{L^\infty (0,\tau ;L^2)}^2+\sigma \Vert \nabla \psi ^{(1)}_t\Vert _{L^2(L^\infty )} \Vert \nabla \overline{\phi }\Vert _{L^\infty (0,\tau ;L^2)}^2\\+\sigma \Vert \nabla \psi ^{(1)}_t\Vert _{L^2(L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,\tau ;L^2)}^2 +\sigma \Vert ^{(2)}\Vert _{L^\infty ( L^\infty )}\Vert \overline{\psi }_t\Vert ^2_{L^2(0,\tau ;L^2)}. \end{multlined}\end{aligned}\) 
Since the functions \(\phi ^{(1)}\)  and \(\phi ^{(2)}\)  belong to \(M_{\textup {K}}\)  and functions \(\psi ^{(1)}\)  and \(\psi ^{(2)}\)  solve the linear problem, we have the uniform bounds
\(\begin{aligned}\Vert \phi _{tt}^{(1)}\Vert _{L^\infty (L^\infty )} \le &\, C_{H^2, L^\infty }R_{\textup {K}}, \\\Vert ^{(2)}\Vert _{L^\infty ( L^\infty )} \le &\, C_{H^2, L^\infty }R_{\textup {K}},\\\Vert \psi _{tt}^{(2)}\Vert _{L^2(L^\infty )} \le &\, C_{H^2, L^\infty }\sqrt{T} C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}}) (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}), \\\Vert \nabla \psi ^{(1)}_t\Vert _{L^2(L^\infty )} \le & \, C_{H^2, L^\infty }\sqrt{T}\Vert \nabla \psi ^{(1)}_t\Vert _{L^\infty (H^2)} \\\le &\, \begin{multlined}[t]C_{H^2, L^\infty }\sqrt{T}\left((1+T+T^{3/2}C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}}))^2 +C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}})^2\right)^{1/2}\\\times (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}), \end{multlined}\end{aligned}\) 
where we have estimated
\(\begin{aligned}\Vert \nabla \psi ^{(1)}_t\Vert _{L^\infty (H^2)}^2\le &\,\Vert \nabla \psi ^{(1)}_t\Vert _{L^\infty (L^2)}^2+\Vert \nabla ^{(1)}_t\Vert _{L^\infty (L^2)}^2\\\le &\, (\Vert \nabla \psi _1\Vert _{L^2}+T\Vert \nabla \psi _2\Vert _{L^2}+T^{3/2}\Vert \nabla \psi ^{(1)}_{ttt}\Vert _{L^2 (L^2)})^2+\Vert \nabla ^{(1)}_t\Vert _{L^\infty (L^2)}^2.\end{aligned}\) 
Therefore, applying Gronwall's inequality to (REF ) results in the bound
\(\begin{aligned}\Vert \overline{\psi }\Vert ^2_{\textup {E}} \le &\, \begin{multlined}[t] C_1(1+T^{3/2}) \, \sqrt{T}\, (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}) \\ \times \exp (C_2 (R_{\textup {K}}+(1+T^{3/2})\sqrt{T}(\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}))\Vert \overline{\phi }\Vert ^2_{\textup {E}}, \end{multlined}\end{aligned}\) 
where the constants \(C_1\) , \(C_2>0\)  do not depend on \(b\)  nor \(T\) . By sufficiently reducing the final time so that
\(\begin{aligned}\begin{multlined}[t]C_1(1+T^{3/2}) \, \sqrt{T}\, (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3})\\ \times \exp (C_2 (R_{\textup {K}}+(1+T^{3/2})\sqrt{T}(\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3})) <1,\end{multlined}\end{aligned}\) 
we can guarantee that
\(\Vert \psi ^{(1)}-\psi ^{(2)}\Vert _{\textup {E}} \le r \Vert \phi ^{(1)}-\phi ^{(2)}\Vert _{\textup {E}} \ \text{ for some } r \in (0,1). \) 
We note that it would be possible to impose smallness of data in the \(H^4 \times H^3\)  norm instead of reducing the final time, analogously to the proof of well-posedness for the Westervelt equation.
We can argue similarly to the proof of Theorem REF  that \((M_{\textup {K}}, d_{\textup {E}})\)  is a closed subset of a complete space with the metric induced by the energy norm \(d_{\textup {E}}(x,y)=\Vert x-y\Vert _{\textup {E}}\) . An application of Banach's Fixed-point theorem to \(\mathcal {T}\)  thus concludes the proof.

The inviscid limit of the Kuznetsov equation
Having obtained the uniform bounds with respect to \(b\)  in the \(X^{\textup {K}}\)  norm, we are now ready to derive the convergence rate for solutions of the Kuznetsov equation as \(b \rightarrow 0^+\) .
Theorem 7.1 
Under the assumptions of Theorem REF , the family of solutions \(\lbrace \psi ^{(b)}\rbrace _{b>0}\)  of the Kuznetsov equation converges in the energy norm to a solution \(\psi \)  of the inviscid Kuznetsov equation at a linear rate. In other words,
\( \Vert \psi ^{(b)}-\psi \Vert _{\textup {E}}\lesssim b \ \mbox{ as } b\rightarrow 0.\) 

The proof follows along the lines of the proof of contractivity in the previous theorem. Let \(b, b^{\prime } \in (0, \overline{b})\) . Let \(\psi ^{(b)}\)  and \(\psi ^{(b^{\prime })}\)  be the solutions of the Kuznetsov equation with the sound diffusivity \(b\)  and \(b^{\prime }\) , respectively, and with homogeneous Dirichlet data, where we assume that initial conditions and final time are chosen according to the well-posedness result of Theorem REF .
The difference \(\overline{\psi }=\psi ^{(b)}-\psi ^{(b^{\prime })}\)  then solves the equation
\( \begin{aligned}&(1-\kappa \psi _t^{(b)} )\overline{\psi }_{tt}-c^2 \Delta \overline{\psi }-b \Delta \overline{\psi }_t\\=&\, \begin{multlined}[t] \kappa \overline{\psi }_t \psi ^{(b^{\prime })} _{tt}+\sigma \nabla \overline{\psi }\cdot \nabla \psi ^{(b)}_t+\sigma \nabla \psi ^{(b^{\prime })} \cdot \nabla \overline{\psi }_t+(b-b^{\prime })\Delta \psi _t^{(b^{\prime })}, \end{multlined}\end{aligned}\) 
supplemented by zero initial conditions. Testing with \(\overline{\psi }_t\)  and integrating with respect to space and time yields
\( \begin{aligned}&\frac{1}{2}\Vert \sqrt{\alpha (t)}\overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert _{L^2(0, t; L^2)}^2 \\=&\,\begin{multlined}[t] -\frac{\kappa }{2} \int _0^t \int _{\Omega }\psi _{tt}^{(b)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(b^{\prime })} \overline{\psi }^2_t \, \textup {d}x\textup {d}s\\+ \sigma \int _0^t \int _{\Omega } (\nabla \overline{\psi }\cdot \nabla \psi ^{(b)}_t+\nabla \psi ^{(b^{\prime })} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s\\+ (b-b^{\prime })\int _0^t \int _{\Omega }_t^{(b^{\prime })} \overline{\psi }_t \, \textup {d}x\textup {d}s, \end{multlined}\end{aligned}\) 
where \(\alpha (t)=1-\kappa \psi ^{(b)}_t(t)\) . We can rely on the following estimate:
\(\begin{aligned}\left|\int _0^t \int _{\Omega } \nabla \overline{\psi }\cdot \nabla \psi ^{(b)}_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right| \le \Vert \nabla \psi ^{(b)}_t\Vert _{L^\infty (0,t ;L^\infty )} \Vert \nabla \overline{\psi }\Vert _{L^2(0,t; L^2)} \Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}.\end{aligned}\) 
Similarly to (REF ), we also have the bound
\(\begin{aligned}\left|\int _0^t \int _{\Omega } (\nabla \psi ^{(b^{\prime })} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s\right|=&\, \left|\frac{1}{2} \int _0^t \int _{\Omega } (- ^{(b^{\prime })}) \overline{\psi }_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right|\\\le &\, \frac{1}{2} \Vert ^{(b^{\prime })}\Vert _{L^\infty (0,t; L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
The two \(\kappa \)  terms on the right-hand side of (REF ) can be estimated as follows:
\(\begin{aligned}&\left|-\frac{1}{2}\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(b)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(b^{\prime })} \overline{\psi }^2_t \, \textup {d}x\textup {d}s\right| \\\le &\, |\kappa | \left(\frac{1}{2} \Vert \psi _{tt}^{(b)}\Vert _{L^\infty (0,t; L^\infty )}+ \Vert \psi _{tt}^{(b^{\prime })}\Vert _{L^\infty (0,t; L^\infty )}\right)\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
Thus, employing the above estimates in (REF ), Young's inequality, and taking the supremum over \(t \in (0, \tau )\)  leads to
\( \begin{aligned}&\frac{1}{2} \underline{\alpha }\sup _{t\in (0,\tau )}\Vert \overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\sup _{t\in (0,\tau )}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert ^2_{L^2(0,\tau ; L^2)} \\\le &\,\begin{multlined}[t] |\kappa | \left(\frac{1}{2}\Vert \psi _{tt}^{(b)}\Vert _{L^\infty (L^\infty )}+\Vert \psi _{tt}^{(b^{\prime })}\Vert _{L^\infty ( L^\infty )}\right)\Vert \overline{\psi }_t\Vert _{L^2(L^2)}^2 \\+ \frac{1}{2}|\sigma |\Vert \nabla \psi ^{(b)}_t\Vert ^2_{L^\infty (L^\infty )} \Vert \nabla \overline{\psi }\Vert _{L^2( L^2)}^2+ \frac{1}{2}|\sigma |\Vert \overline{\psi }_t\Vert ^2_{L^2( L^2)}\\+\frac{1}{2}|\sigma |\Vert ^{(b^{\prime })}\Vert _{L^\infty (L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2( L^2)}^2 + \frac{1}{2} (b-b^{\prime })^2 \Vert _t^{(b^{\prime })}\Vert _{L^2( L^2)}^2+\frac{1}{2}\Vert \overline{\psi }_t\Vert _{L^2( L^2)}^2. \end{multlined}\end{aligned}\) 
By virtue of Theorem REF , we have the uniform bound
\(\begin{aligned}\begin{multlined}[t] \Vert \psi _{tt}^{(b)}\Vert _{L^\infty (L^\infty )}+\Vert \psi _{tt}^{(b^{\prime })}\Vert _{L^\infty (L^\infty )}+\Vert \nabla \psi ^{(b)}_t\Vert ^2_{L^\infty (L^\infty )} \\+\Vert ^{(b^{\prime })}\Vert _{L^\infty (L^\infty )} \lesssim R_{\textup {K}} + R_{\textup {K}}^2, \end{multlined}\end{aligned}\) 
and \(\Vert _t^{(b^{\prime })}\Vert _{L^2 (L^2)}^2 \lesssim T R_{\textup {K}}^2\) . Therefore applying Gronwall's inequality results in
\(\Vert \overline{\psi }\Vert _{\textup {E}}=\Vert \psi ^{(b)}-\psi ^{(b^{\prime })}\Vert _{\textup {E}} \lesssim |b-b^{\prime }|.\) 
Finally, we note that the estimate remains valid for \(b^{\prime }=0\)  and thus (REF ) holds, as claimed.

Conclusion
In this work, we have performed a convergence study of parabolic approximations to quasilinear wave equations of second order as the damping parameter \(b\)  tends to zero. Our research has been motivated by two classical nonlinear acoustics models, the Westervelt and Kuznetsov equations. As the analysis showed, the solutions of these equations converge in the energy norm with a linear rate to the solutions of the respective inviscid equations (with \(b=0\) ), provided that the initial data are smooth and small and the final time short enough. For a short enough final time, the smallness condition on the initial data can be imposed in a lower-order space than the one needed to derive higher-energy bounds. In particular, the smallness condition on the data only arises from ensuring that the non-degeneracy assumption on \(p\)  and \(\psi _t\)  holds in the Westervelt and Kuznetsov equations, respectively.

Acknowledgment
The work of the first author was supported by the Austrian Science Fund fwf under the grants P30054 and DOC 78.
